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THE   FOURTH   DIMENSION 
AND  THE  BIBLE 

BY  ^ 

WILLIAM  ANTHONY  GRANVILLE,  Ph.D., 
LL.D> 

PRESIDENT  OF  GETTYSBURG  COLLEGE. 

Formerly   Instructor   of  Mathematics  in    Yale    University;   Author  of 
Granville's    Differential    and    Integral    Calculus,    Granville's    Plane 
Trigonometry,    Granville's    Spherical    Trigonometry,    and    Gran- 
ville's Logarithmic  Tables;  loint  Author  of  Smith  and  Gran- 
ville's  Elements    of   Analysis;   Inventor    of   Polar    Co- 
ordinate Plotting  Paper,  Yale  System  of  Mathemat- 
ical Note  Books,  and  Granville' s   Transparent 
Combined   Ruler   and   Protractor. 


BOSTON 
RICHARD  G.  BADGER 

THE   GORHAM    PRESS 


Copyright,  1922,  by  William  A.  Granville 


All  Rights  Reserved 

.^7 


Made  in  the  United  States  of  America 


The  Gorham  Press,  Boston,  U.  S.  A. 

J30STON  mUM^E  MBRARl 

r1AY  11  1972 
477046 


PREFACE 

That  philosophy  and  the  physical  sciences, 
when  called  to  the  defence  of  Christianity,  have 
so  often  proven  to  be  broken  reeds  has  been  a 
keen  disappointment  to  many  sincere  seekers  after 
the  truth.  A  little  reflection,  however,  will  make 
clear  that  in  the  very  nature  of  things  such  fail- 
ures must  inevitably  result,  because  structures 
resting  on  shifting  sand  can  never  serve  as  strong 
buttresses.  No  two  systems  of  philosophy  agree 
and  every  philosopher  is  continually  tinkering 
with  his  own  system.  There  is  no  final  authority 
in  philosophy  to  which  an  appeal  can  be  made. 
Such  being  the  case  it  is  obviously  not  to  be  ex- 
pected that  philosophy  can  furnish  us  with  proofs 
of  the  truths  of  Christianity  whose  validity  will  be 
generally  recognized.  And  the  sciences  have  also 
fallen  short  for  similar  reasons.  The  foundations 
of  our  physical  sciences  are  far  from  stable.  To 
doubt  the  principles  of  the  conservation  of  energy, 
the  conservation  of  matter,  or  the  Indivisibility  of 
the  atom,  would  have  been  considered  as  rank 
scientific  heresy  a  few  years  ago;  today  all  physi- 
cists are  familiar  with  phenomena  which  contradict 


iv  Preface 

one  or  more  of  them.  The  Einstein  Theory  of 
Relativity  is  now  opening  up  new  fields  for  scien- 
tific investigation  which  promises  results  that  will 
practically  revolutionize  many  current  fundamen- 
tal scientific  concepts.  In  fact,  the  foundation 
principles  on  which  the  physical  sciences  rest  have 
been  changing  so  rapidly  of  late  that  practically 
all  the  science  text-books  now  used  in  our  schools 
are  more  or  less  obsolete.  The  physical  sciences 
as  well  as  philosophy  are  in  a  constant  state  of 
flux  and  flow  as  well  as  of  growth,  and  this  must 
necessarily  continue  to  be  the  situation  until  the 
ultimate  truth  is  reached,  If  it  is  ever  attained,  in 
these  domains  of  knowledge.  As  all  will  readily 
agree  that  the  limits  of  these  regions  have  not  yet 
been  reached  and,  what  is  more,  that  these  limits 
are  not  even  measurably  in  sight,  it  is  evident 
that  we  cannot  reasonably  expect  that  much  con- 
structive light  will  be  thrown  on  Christianity  now 
or  in  the  near  future  by  either  philosophy  or  the 
physical  sciences. 

To  satisfy  completely  human  reason  any  confir- 
mation of  our  Christian  beliefs  must  come  from  a 
source  whose  authority  no  one  can  question.  Is 
there  then  any  department  of  human  knowledge 
whose  foundations  rest,  not  on  shifting  sand,  but 
on  the  bed  rock  of  absolute  truth?  Pure  mathe- 
matics satisfies  this  condition ;  it  is  the  only  exact 
science  that  God  has  revealed  to  man  and  the 


Preface  v 

truths  which  it  contains  are  the  only  truths  that 
can  be  absolutely  established  thru  pure  reason. 
Because  pure  mathematics  reveals  absolute  truth 
it  is  part  of  God  Himself,  for  God  is  the  essence 
of  all  truth.  It  was  Emerson  who  said,  "nature 
geometrizes,"  and  as  nature  is  the  handiwork  of 
God  we  may  say  that  God  geometrizes.  In  view 
of  this  it  is  rather  surprising  that  so  few  attempts 
should  have  been  made  in  the  past  to  throw  light 
from  mathematics  on  the  fundamentals  of  Chris- 
tianity. This  is  practically  a  virgin  field  for 
theological  research. 

Pure  mathematics  is  the  vanguard  of  all  the 
sciences,  it  has  marched  centuries  in  advance  of 
them  all.  For  instance,  Calculus  was  discovered 
nearly  two  hundred  years  before  the  science  of 
Chemistry  was  far  enough  advanced  to  allow  the 
use  of  Calculus  as  one  of  its  instruments  of  re- 
search. Practically  the  same  thing  is  true  of  all  the 
other  physical  sciences,  including  engineering.  It 
may  be,  therefore,  that  the  science  of  theology  is 
not  yet  far  enough  advanced  to  admit  of  the  ready 
use  of  mathematics  as  an  instrument  of  research. 
Today  mathematicians  are  forging  ahead  into 
realms  of  abstract  thought  and  pure  logic  as  never 
before;  judging  from  the  past,  some  of  the  truths 
which  they  are  revealing  now  may  not  be  put  to 
what  we  are  pleased  to  call  practical  use  for 
centuries  to  come. 


vi  Preface 

That  mathematics  will  ultimately  prove  to  be 
a  valuable  aid  in  solving  many  of  the  perplexing 
problems  connected  with  our  Christian  religion  is 
the  firm  belief  of  many  sound  thinkers.  Professor 
C.  J.  Keyser  has  pointed  out  many  striking  cor- 
respondences between  the  transcendental  proper- 
ties of  some  elementary  mathematical  concepts 
and  those  transcendental  attributes  of  God  and 
His  creation  which  are  intimately  associated  with 
our  religious  behefs.  Concerning  the  possibility 
and  the  probability  of  future  investigations  along 
this  and  other  lines  shedding  new  light  from 
mathematical  sources  on  current  religious  thought 
he  has  the  following  to  say  in  Hibhert  Journal, 
Vol.  VII :  "Over  and  above  the  humbler  role  of 
mathematics  as  a  metrical  and  computatory  art, 
over  and  above  her  unrivalled  value  as  a  standard 
of  exactitude  and  as  an  instrument  in  every  field 
of  experimental  and  observational  research,  even 
beyond  her  justly  famed  disciplinary  and  emanci- 
pating power,  releasing  the  faculties  from  the 
fickle  dominion  of  sense  by  winning  their  allegiance 
to  the  things  of  the  spirit,  inuring  them  to  the 
austerities  of  reason,  the  stern  demands  of  rigor- 
ous thought,  giving  the  mental  enlargement,  the 
peaceful  perspective,  the  poise  and  the  elevation 
that  come  at  length  from  continued  contemplation 
of  the  universe  under  the  aspects  of  the  infinite 
and  the  eternal — my  conviction  that  above  and 


Preface  vii 

beyond  these  services,  which  by  common  consent 
of  the  competent  are  peculiarly  her  own,  mathe- 
matics will  yet  further  demonstrate  her  human 
significance  by  the  shedding  of  light  more  and 
more  copious  as  the  years  go  by  on  ultimate  prob- 
lems of  philosophy  and  theology,  is  not  a  passing 
fancy,  or  a  momentary  whim." 

The  only  motive  of  the  author  in  writing  this 
book  is  his  sincere  desire  to  throw  some  light, 
even  though  very  dim,  on  some  of  the  questions 
connected  with  our  Christian  beliefs.  This  is  not, 
however,  an  endeavor  to  construct  a  system  of 
Christian  theology  on  mathematical  foundations. 
The  author's  chief  aim  will  be  to  point  out  the 
remarkable  agreement  which  exists  between  nu- 
merous Bible  passages  and  some  of  the  concepts 
which  follow  quite  naturally  from  the  mathemat- 
ical hypothesis  of  higher  spaces.  In  making  this 
attempt  the  author  is  well  aware  that  he  is  on  a 
"no  man's  land"  exposed  to  fire  from  the  mathe- 
matical trenches  on  the  one  side  and  the  theological 
trenches  on  the  other.  But  "fools  step  in  where 
angels  fear  to  tread"  and  sometimes  they  survive 
to  tell  the  tale. 

The  scripture  passages  quoted  in  this  text  are 
not  employed  for  exegetical  purposes  but  to  show 
that  the  conclusions  arrived  at  in  no  way  con- 
tradict the  divine  word. 

While  there  is  quite  an  extensive  bibliography 


vill  Preface 

dealing  with  the  subject  of  higher  spaces,  the  liter- 
ature bearing  on  their  relation  to  Bible  state- 
ments is  practically  nil.  After  a  thoro  search  only 
scattered  references,  each  embracing  only  a  few 
phrases  or  sentences,  have  been  found  which 
might  be  construed  as  supporting  some  of  the 
conclusions  arrived  at  in  this  text.  The  author 
is  therefore  taking  a  voyage  on  practically  un- 
charted seas.  No  doubt  it  may  seem  presump- 
tuous on  the  part  of  the  author,  who  aspires  to 
be  a  mathematician  and  is  no  theologian  at  all, 
to  poach  thus  on  theological  preserves;  but  the 
case  is  perhaps  no  worse  than  if  a  theologian  who 
is  not  also  a  mathematician  should  undertake  the 
task. 

A  sincere  effort  has  been  made  to  present  the 
matter  in  popular  form.  To  understand  the  text  it 
is  not  necessary  for  the  reader  to  be  either  a  pro- 
fessional mathematician  or  a  profound  theologian. 
No  extended  excursion  into  the  theory  of  higher 
spaces  will  be  attempted;  the  author  has  limited 
himself  to  those  fundamental  concepts  only  which 
are  necessary  for  the  illumination  of  the  applica- 
tions to  be  made.  No  proofs  will  be  given  of  the 
mathematical  propositions  hereinafter  stated;  but 
the  reader  can  rely  on  them  absolutely,  no  matter 
how  startling  the  conclusions  may  appear.  They 
have  all  been  proven  by  mathematicians  of  the 
highest  repute  and  the  validity  of  these  proofs 


Preface  ix 

are  not  more  in  dispute  than  is  the  proposition 
that  2X2  =  4. 

The  use  of  the  term  "Fourth  Dimension"  as 
part  of  the  title  of  this  book  does  not  mean  that 
we  shall  confine  our  discussion  to  that  space  ex- 
clusively, important  as  it  is ;  it  is  so  used  because 
in  the  populai^  mind  this  term  represents  the 
mathematical  concept  of  higher  spaces  in  general. 

William  Anthony  Granville. 

Gettysburg  College, 

Gettysburg,  Pa. 


CONTENTS 

CHAPTER  PAGE 

I    The  Concept  of  Space ii 

II     Geometric  Units  in  Each  Space i8 

III  On  the  Existence  of  Higher  Spaces 24 

IV  About  the  Supposed  Inhabitants  of  Other  Spaces 

Than  Our  Own.     Flatland.     Lineland     .     .  36 

V    Motion  of  a  Point  Perpendicular  to  Each  Space  47 

VI    Rotation  of  Symmetrical  Configurations     ...  54 

VII     Slctions  of  Bodies  in  One  Space  Made  by  Spaces 
OF  a  Lower  Order.    Conservation  of  Matter 

and    Energy 69 

VIII    The  Illusory  and  the  Real 82 

IX    Spaces  as  Heavens  and  Hells 86 

X    Creation  of  Our  Material  Universe.     Evolution  91 

XI    Raising  of  the   Dead.    Sacrament  of  the   Lord's 

Supper 99 

XII     Description  of   Heaven.     Miracles 106 

XIII     Conclusion        no 


THE  FOURTH  DIMENSION 
AND  THE  BIBLE 


THE   FOURTH   DIMENSION 
AND   THE   BIBLE 

CHAPTER   I 

THE  CONCEPT  OF  SPACE 
(a.)  Zero-Dimensional  Spaces.    A* 

A  point,  as  A,  may  be  conceived  of  as  a  space 
of  no  dimensions.  Every  point  in  that  space 
coincides  with  A  and  therefore  is  at  no  distance 
from  it.  A  point  cannot  move  about  in  that 
space,  hence  its  degree  of  freedom  of  motion  is 
zero.  Since  we  know  that  there  is  an  infinite 
number  of  points  there  is  also  an  infinite  number 
of  such  zero-dimensional  spaces. 

(b)    One-Dimensional  Spaces. 


Let  O  be  a  fixed  point  on  a  straight  line.  To 
determine  the  position  of  any  point  on  that  line, 
as  A,  all  we  need  to  know  is  its  distance  to  the 


12     The  Fourth  Dimension  and  the  Bible 

right  or  left  of  O  expressed  in  terms  of  some  unit 
of  length,  a  positive  number  to  indicate  that  it 
is  so  far,  say,  to  the  right  of  O,  or  a  negative 
number  when  it  is  to  the  left  of  O.  A  single  num- 
ber will  therefore  suffice  to  determine  the  posi- 
tion of  any  point  on  the  straight  line;  hence  the 
straight  line  is  said  to  be  one-dimensional.  The 
straight  line  may  also  be  conceived  of  as  a  space 
of  one  dimension;  any  point  in  it  has  only  one 
degree  of  freedom  of  motion,  it  can  move  only 
back  and  forth  in  the  line. 


The  circumference  of  a  circle  may  be  consid- 
ered as  a  curved  one-dimensional  space,  as  well 
as  every  other  curved  line,  whether  it  returns  into 
itself  or  not,  because  (as  in  the  case  of  the 
straight  line)  the  position  of  any  point,  as  A,  may 
be  determined  by  a  single  number  representing 
its  distance  measured  along  the  curve  from  a  fixed 
point  (as  O  •)  on  that  curve.  Obviously  there  is 
an  infinite  number  of  one-dimensional  spaces  be- 
cause the  number  of  straight  and  curved  lines  is 
infinite. 


The  Concept  of  Space 


13 


(c)    Two-Dimensional  Spaces. 

y. 

B 


0 


Let  us  now  consider  the  plane  of  the  two  mutu- 
ally perpendicular  lines  OX  and  OY  intersecting 
at  O.  Any  point  in  this  plane,  as  A,  may  be 
determined  by  its  perpendicular  distances  BA  and 
CA  from  OY  and  OX.  Calling  distances  from 
OY  and  OX  measured  to  the  right  and  upward 
as  positive,  and  distances  to  the  left  and  down- 
ward as  negative,  the  position  of  any  point  in  the 
plane  will  be  determined  by  the  two  numbers  rep- 
resenting these  distances;  hence  the  plane  is  said 
to  be  two-dimensional.  This  plane  may  also  be 
conceived  of  as  a  space  of  two  dimensions;  any 
point  in  it  has  two  degrees  of  freedom  of  motion, 
it  can  move  to  the  right  or  left,  and  up  or  down. 
A  point  in  the  plane  can  move  from  one  position 
to  any  other  in  that  plane  by  moving  in  a  path 
which  is  always  parallel  to  one  of  two  fixed 
mutually  perpendicular  lines  in  that  plane.  Take, 
for  instance,  the  top  of  a  rectangular  table.    Any 


14     The  Fourth  Dimension  and  the  Bible 

article  on  it,  as  an  Ink  stand,  may  be  moved  from 
one  spot  to  any  other  along  a  path  that  will  always 
be  parallel  to  one  of  the  edges  of  the  table  top. 

The  surface  of  a  sphere  may  be  conceived  of  as 
a  curved  two-dimensional  space.  The  position  of 
any  point  on  the  surface  of  the  earth  Is  deter- 
mined when  we  know  the  two  numbers  indicating 
its  geographical  latitude  and  longitude.  By  mov- 
ing along  parallels  of  latitude  and  lines  of 
longitude  we  may  travel  from  any  one  place  to 
any  other.  Not  only  the  surfaces  of  all  spheres 
but  also  an  unlimited  number  of  other  curved 
surfaces  may  be  conceived  of  as  curved  two- 
dimensional  spaces.  Evidently  there  Is  an  Infinite 
number  of  two-dimensional  spaces. 

(d)    Three-Dimensional  Spaces. 


f^^f 


The  Concept  of  Space  15 

The  space  in  which  we  move  and  have  our  be- 
ing is  three-dimensional ;  we  say  that  bodies  in  it 
have  length,  breadth  and  thickness.  Consider 
three  mutually  perpendicular  planes  meeting  at  O. 
The  position  of  any  point  in  our  space,  as  A,  is 
then  determined  by  the  three  numbers  represent- 
ing its  perpendicular  distances  BA,  CA,  DA,  with 
the  proper  signs,  from  the  three  planes.  A  point 
in  our  space  has  three  degrees  of  freedom,  it 
can  move  to  the  right  or  left,  forward  or  back- 
ward, and  up  or  down.  For  instance,  an  article 
in  a  rectangular  shaped  room  may  be  moved  from 
one  place  to  any  other  by  moving  it  parallel  to 
the  directions  of  the  length,  breadth,  and  height 
of  the  room.  The  reader  can  reach  no  place 
which  may  not  be  reached  by  going  north  or 
south,  east  or  west,  and  upward  or  downward. 

The  space  we  live  in  is  the  only  three-dimen- 
sional space  of  which  we  have  any  knowledge 
gained  by  experience.  We  are  not  warranted, 
however,  in  making  the  statement  that  no  other 
three-dimensional  spaces  exist;  such  a  proposition 
cannot  be  proven.  Curved  lines  have  given  us 
the  conception  of  curved  one-dimensional  spaces, 
and  curved  surfaces  the  conception  of  curved  two- 
dimensional  spaces.  We  may  also  conceive  of 
the  existence  of  curved  three-dimensional  spaces 
although  in  this  case  we  have  no  corresponding 


1 6     The  Fourth  Dimension  and  the  Bible 

visual  Images  on  which  to  lean;  we  cannot  con- 
struct in  our  minds  the  picture  of  a  curved  three- 
dimensional  space.  From  analogy  it  is  evident 
that  there  may  be  an  infinite  number  of  three- 
dimensional  spaces,  both  curved  and  not  curved. 

(e)  Spaces  of  Order  Higher  than  the  Third, 
also  called  Hyper-Spaces. 

In  like  manner  we  define  a  four-dimensional 
space  to  be  one  in  which  it  requires  four  numbers 
to  determine  the  position  of  any  point.  A  point 
in  such  a  space  will  have  four  degrees  of  free- 
dom. 

A  five-dimensional  space  would  be  one  in  which 
it  requires  five  numbers  to  determine  the  position 
of  a  point;  a  point  in  it  would  have  five  degrees 
of  freedom.  Similarly  we  define  spaces  of  6,  7, 
8,  .   .   .  n,  dimensions. 

It  is  impossible  for  us  to  form  a  mental  picture 
of,  say,  the  fourth  dimension,  the  idea  is  wholly 
intangible.  Nevertheless,  it  is  not  meaningless 
or  an  absurdity,  but  a  useful  mathematical  con- 
cept which  has  led  to  the  development  of  a  geom- 
etry of  four  dimensions  Involving  no  contradic- 
tions. We  reach  this  new  concept  of  higher 
spaces  not  at  a  single  leap  but  slowly  and  gradu- 
ally by  climbing  a  ladder  whose  lowest  rungs  are 
the  lower  dimensional  spaces  with  which  we  are 


The  Concept  of  Space  17 

in  a  certain  sense  familiar.  Guided  by  analogy 
with  dimensions  of  a  lower  order  we  shall  thus 
try  to  arrive  step  by  step  to  at  least  a  partial  and 
symbolic  idea  of  the  meaning  of  higher  dimen- 
sions. Of  these  we  will  be  chiefly  concerned  with 
the  fourth  dimension.  Some  claim  that  the  Bible 
recognizes  space  of  four  dimensions  in  Job 
11:8,9  ^"d  Eph.  3:18  where  length,  breadth, 
depth  and  height,  i.  e.,  four  dimensions,  are  men- 
tioned. 


CHAPTER   II 

GEOMETRIC  UNITS  IN  EACH  SPACE 

That  there  exist  intimate  relations  closely  ap- 
proaching a  sort  of  interdependence  between  the 
spaces  of  various  orders  is  forcibly  suggested  by 
the  following  discussion  showing  how  the  unit  in 
each  space  may  be  generated  by  the  unit  of  the 
space  of  the  next  lower  order  moving  in  a  new 
direction  not  contained  within  itself. 

Starting  with  zero-dimensional  space  (a  point) 
we  may  consider  the  point  itself  as  the  unit. 

•A  POINT 

Zero-dimensional 
Space   Unit 

If  the  point  A  moves  the  distance  of,  say,  one 
inch  to  the  right  it  will  generate  the  unit  line  AB. 

LINE 

,         ,  Bounded    by    the    points    A    and    B 

One-dimensional   Space  (zero-dimensional    units) 

Unit 

If  the  line  AB  moves  upward  one  Inch  In  a 
i8 


Geometric  Units  in  Each  Space 


19 


direction  perpendicular  to  itself  it  will  generate 
the  unit  square  ABCD. 

SQUARE 


Bounded  by  the  four  unit  lines 
AB,  BC,  CD,  DA. 

(one-dimensional  units) 


Two-dimensional  Space  Unit 

If  the  square  ABCD  moves  backward  one  inch 
in  a  direction  perpendicular  to  itself  it  will  gener- 
ate the  unit  cube  ABCDEFGH. 


CUBE 


Bounded  by  the  six  unit 
squares  ABCD,  EFGH, 
ABFE,  DCGH,  AEHD, 
BFGC  •  (two-dimensional 
units). 


Three-dimensional  Space  Unit 

If  the  cube  ABCDEFGH  moves  away  one  inch 
in  a  direction  perpendicular  to  itself  it  will  gener- 
ate the   corresponding  unit   in   four-dimensional 


20     The  Fourth  Dimension  and  the  Bible 

space.  This  unit  has  been  given  the  name  tesser- 
act.  Here  we  meet  our  first  serious  difficulty. 
We  are  familiar  with  the  idea  of  motion  perpen- 
dicular to  a  line,  and  of  motion  perpendicular  to 
a  square  (or,  what  amounts  to  the  same  thing,  of 
motion  perpendicular  to  two  mutually  perpendicu- 
lar lines  lying  in  the  plane  of  the  square),  but  we 
cannot  conceive  of  a  motion  perpendicular  to  a 
cube  (i.  e.,  a  single  motion  perpendicular  to  all 
the  faces  of  the  cube,  or,  as  it  may  also  be  stated, 
perpendicular  to  any  three  adjacent  edges  of  the 
cube).  Not  one  of  our  senses  calls  for  a  fourth 
direction  perpendicular  to  the  other  three,  expe- 
rience leaves  us  satisfied  with  three  dimensions. 
We  cannot  visualize  the  tesseract  but  it  is  possible 
for  us  to  construct  a  figure  in  our  space  which  will 
symbolize  the  tesseract,  that  is,  a  figure  having 
many  of  the  properties  of  the  tesseract.  Such  a 
figure  is  the  following  which  we  may  conceive  of 
as  having  been  generated  by  the  original  cube 
ABCDEFGH  expanding  at  a  uniform  rate  into 
the  cube  A'B'CD'ET'G'H^  the  directions  AA', 
BB',  CC,  DD^  EE^  FF^  GG^  HH',  being  sup- 
posed perpendicular  to  our  space  and  equal  in 
length  to  an  edge  of  the  cube. 

In  generating  a  tesseract  by  the  motion  of  a 
cube,  the  latter's  corners  generate  edges,  its  edges 
generate  faces   (squares)   and  its  faces  generate 


Geometric  Units  in  Each  Space  21 

TESSERACT 

(Symbolical) 


D 


/i' 


Bounded  by  the  eight  unit 
cubes  ABCDEFGH,  A'B' 
C'D'E'F'G'H',  A'B'BAE' 
F'FE,  DCC'D'HGG'H', 

BB'C'CFF'G'G,  A'ADD' 
E'EHH'  A'B'C'D'ABCD, 
EFGHE'F'G'H'  (three-di- 
mensional   units). 


Four-dimensional    Space    Unit 

cubes.  The  faces  (squares)  generated,  ABB'A', 
BFF'B',DCCD',  etc.,  and  the  cubes  generated, 
A'B'BAE'FTE,  BBT'CFF^G^G,  etc.,  necessarily 
appear  distorted  in  the  above  figure.  The  num- 
ber of  the  various  elements  of  the  tesseract  are 
indicated  in  the  following  table  which  may  be 
easily  verified  by  the  reader. 

Number  in        Number      Number  in    Number  in 
Initial  Cube.  Generated.  Final  Cube.  Tesseract. 

Corners    8         +  o         +  8         =         i6 

Edges    13         +  8         +         12        =        32 

Faces    6        +         i3        +  6        =        24 

Cubes    I        +  6        +  I        =  8 


The  statements  made  opposite  the  figures  on 
pp.  18,  19,  20  indicate  that  each  space  unit  is 
bounded  by  space  units  of  the  next  lower  order. 


22     The  Fourth  Dimension  and  the  Bible 

The  following  table  indicates  the  number  of  each 
of  the  various  elements  that  are  associated  with 
the  boundaries  of  the  various  space  units. 

Points.  Lines.  Squares.  Cubes. 

Zero-dimensional  unit   (point)    . ,      i  o            o            o 

One-dimensional  unit   (line)    ....      3  1            o            o 

Two-dimensional  unit   (square)    .4  4             i             o 

Three-dimensional   unit    (cube)    .       8  12             6             i 

Four-dimensional  unit   (tesseract)     16  32          24            8 

Since  there  is  an  infinite  number  of  points  in  a 
line,  an  infinite  number  of  lines  in  a  square,  and 
an  infinite  number  of  squares  in  a  cube,  it  is  near 
at  hand  to  assume  that  a  space  of  any  order  con- 
tains an  infinite  number  of  lower  dimensional 
spaces.  The  following  definition  of  space  has 
therefore  been  suggested: 

Space  is  that  which  separates  two  portions  of 
the  next  higher  space  from  each  other. 

Let  us  illustrate  what  is  meant. 

The  point  A  (zero-dimensional 
space)  divides  the  line  (one- 
dimensional  space)  into  two  parts. 


The     line     AB      (one-dimensional 
space)  divides  the  plane  CE  (two- 
R      dimensional  space)   into  two  parts. 


D 


Geometric  Units  in  Each  Space  23 

d 


The  plane  AB  (two-dimen- 
sional space)  divides  the  solid 
CD  (three-dimensional  space) 
into  two  parts. 


Analogy  now  steps  in  and  suggests  that  we  carry 
this  one  step  further  by  making  the  statement  that 
our  three-dimensional  space  probably  divides 
four-dimensional  space  into  two  parts.  And  so  on 
for  the  other  higher-dimensional  spaces. 


CHAPTER   III 

ON  THE  EXISTENCE  OF  HIGHER 
SPACES 

It  is  to  be  hoped  that  by  this  time  there  has 
dawned  in  the  mind  of  the  reader  at  least  a  faint 
idea  of  the  concept  of  higher  spaces,  an  inkling 
of  what  is  meant  by,  say,  the  fourth  dimension. 
Additional  discussions  from  other  viewpoints 
tending  to  show  the  probability  of  the  existence 
of  space  of  four  dimensions  will  be  given  in  later 
chapters  of  this  book. 

We  experience  no  difficulty  in  forming  a  geo- 
metric conception  of  zero-dimensional  space 
(point),  of  one-dimensional  space  (line),  of  two- 
dimensional  space  (plane),  and  of  three-dimen- 
sional space  (our  universe)  ;  but  when  at  first  we 
try  to  give  a  geometric  interpretation  to  space  of 
four,  five,  six  and  higher  dimensions,  we  are  at 
sea.  We  find  nothing  tangible  on  which  to  lean 
in  our  fund  of  experiences  accumulated  in  dealing 
with  the  material  things  in  our  world,  things  of 
which  we  have  become  conscious  thru  our 
senses.     We  are  therefore  tempted  to  at  once 

24 


On  the  Existence  of  Higher  Spaces       25 

arrive  at  the  conclusion  that  no  spaces  of  four, 
five,  six,  etc.,  dimensions  exist.  Such  a  conclusion, 
however,  would  be  premature  because,  as  has  al- 
ready been  stated,  the  proposition  that  higher 
spaces  do  not  exist  cannot  be  proven. 

Firstly,  to  dismiss  the  concept  of  the  existence 
of  higher-dimensional  spaces  with  the  statement 
that  such  a  thing  is  incomprehensible  is  begging 
the  question,  unless  we  are  also  willing  to  abandon 
the  idea  of  the  existence  of  our  own  space  of  three 
dimensions,  because  Its  properties  are  also  In  the 
main  unfathomable  mysteries  to  us.  For  in- 
stance, "It  may  properly  be  claimed  that  a  three- 
dimensional  solid  of  infinite  length,  infinite 
breadth,  and  Infinite  thickness  would  embrace  in- 
finite space;  but  is  it  possible  to  picture  or  com- 
prehend what  infinite  space  Is?  Can  a  finite  mind 
picture  a  space  with  no  beginning  and  no  ending; 
limitless  space  In  which  our  vast  solar  system  is 
a  mere  dot,  In  which  the  known  stellar  universe 
Is  probably  also  comparatively  a  mere  dot,  al- 
tho  it  is  actually  so  vast  in  extent  that  the 
light  from  some  of  Its  component  stars  which 
started  towards  us  generations  ago  or  centuries 
ago  is  only  now  reaching  us?  If  our  space  is 
limitless,  the  idea  is  Incomprehensible,  and  if  it 
is  limited,  its  limits  are  Incomprehensible.  Our 
space  is  limited  or  it  is  limitless;  in  either  case 


26     The  Fourth  Dimension  and  the  Bible 

the  idea  is  incomprehensible.  Thus  the  mere 
statement  that  a  concept  is  incomprehensible  does 
not  prove  its  non-existence."  (Gunnell.)  Obvi- 
ously, it  would  be  absurd  for  us  to  doubt  the  ex- 
istence of  our  own  space  simply  because  many  of 
its  properties  are  incomprehensible  to  us.  The 
human  mind  cannot  fully  encompass  the  idea  of 
infinity  and  yet  this  is  one  of  the  most  useful 
mathematical  concepts  employed  daily  without 
hesitation  in  the  teaching  of  elementary  as  well 
as  advanced  mathematics. 

Secondly,  only  an  ignoramus  would  claim  that 
nothing  exists  which  does  not  lie  within  his  realm 
■of  understanding,  knowledge  and  experience. 
One  hundred  years  ago  the  results  now  achieved 
thru  the  telephone,  the  phonograph,  or  the 
wireless  telegraph  and  telephone  would  have  been 
inconceivable,  would  have  appeared  as  miracles 
even  to  the  wisest  men  of  that  day. 

Thirdly,  probably  few  readers  of  this  book 
would  hesitate  to  make  the  assertion  that  one- 
dimensional  spaces  (lines)  and  two-dimensional 
spaces  (planes)  do  actually  exist.  But  as  a  mat- 
ter of  fact,  the  line  as  a  one-dimensional  space 
and  the  plane  as  a  two-dimensional  space  has  no 
more  of  a  real  existence  to  us  than  has  four- 
dimensional  space.  No  one  has  ever  seen  or 
heard  or  felt  or  tasted  or  smelled  either  a  line 


On  the  Existence  of  Higher  Spaces       27 

or  a  plane.  All  our  sense  perceptions  and  the  ex- 
periences due  to  them  are  three-dimensional  be- 
cause all  material  things  are  three-dimensional. 
Our  senses  can  perceive  only  material  things  and 
those  only  thru  our  sensory  nerves  which  are 
also  material  and  therefore  three-dimensional. 
Even  the  shadow  on  a  wall  cannot  truly  be  said 
to  be  two-dimensional  because  what  we  really  ob- 
serve is  that  fewer  rays  of  light  enter  our  eyes 
from  the  region  covered  by  the  shadow,  while 
from  the  region  surrounding  the  shadow  many 
rays  of  light  do  enter  our  eyes.  And  these  rays 
of  light  are  reflected  from  material  particles 
(pigment)  on  the  surface  of  the  wall,  that  is, 
from  particles  of  matter  (three-dimensional). 
Our  eyes  communicate  to  our  brains  only  three- 
dimensional  knowledge;  ''for  the  comprehension 
of  anything  which  has  only  one  or  two  dimensions 
a  purely  intellectual  act  of  abstraction  must  he 
added  to  that  of  perception."  Lines  drawn  on 
paper,  strings,  rods  and  other  objects  which  sug- 
gest to  us  the  geometric  idea  of  a  line  are  really 
three-dimensional  forms  of  matter  in  which  two 
dimensions  are  very  small  as  compared  with  the 
third,  its  length.  Similarly  the  plane  surfaces  of 
our  experience  (sheets  of  paper,  blackboard, 
etc.)  are  all  three-dimensional  forms  in  which  the 
thickness   is  very  small  a§   compared  with  the 


28     The  Fourth  Dimension  and  the  Bible 

length  and  breadth;  but  these  helps  have  been 
found  useful  to  us  in  visualizing  the  geometric 
idea  of  a  plane.  Thus  lines  drawn  on  paper  or 
on  a  blackboard  have  been  found  very  helpful  in 
teaching  plane  geometry,  but  they  are  not  a  part 
of  plane  geometry  itself.  If  all  such  material  aids 
were  discarded,  or  should  cease  to  exist,  plane 
geometry  as  we  know  it  would  still  remain,  and 
every  theorem  proven  in  the  plane  geometry  text- 
books used  in  our  schools  today  would  still  hold 
true.  Four-dimensional  geometry  exists  in  the 
same  sense  that  plane  geometry  exists,  both  have 
a  real  existence.  The  difficulty  we  encounter  in 
our  attempts  to  form  mental  images  of  geometric 
figures  of  four  dimensions  consists  in  the  fact 
that  we  have  no  material  aids  to  our  imagination, 
while  figures  in  plane  and  solid  geometry  are 
readily  conceived  of  as  corresponding  to  many 
material  objects  in  the  world  with  which  we  are 
familiar. 

Geometry  itself,  therefore,  throws  no  light  on 
the  real  nature  of  any  space,  not  even  on  the 
nature  of  the  three-dimensional  space  in  which 
we  live.  In  fact,  it  has  not  even  been  proven 
that  the  solid  geometry  taught  in  our  schools 
and  the  space  in  which  we  live  are  in  exact  cor- 
respondence; all  we  can  say  is  that  as  far  as  our 
observations  have  extended,  the  one  does  appear 


On  the  Existence  of  Higher  Spaces       29 

to  fit  the  other.  The  study  of  space  is  an 
empirical  science,  its  conclusions  must  be  ar- 
rived at  thru  observation  and  experiments,  while 
geometry  is  a  science  built  up  by  pure  reason,  it 
is  a  branch  of  pure  mathematics.  To  the  mathe- 
matician the  geometry  of  four  dimensions,  as  well 
as  the  geometry  of  two  dimensions,  has  just  as 
real  an  existence  as  the  geometry  of  three  dimen- 
sions, the  geometry  relating  to  the  space  of  our 
experience.  Many  extended  treatises  have  been 
written  on  the  geometry  of  higher  spaces  by 
mathematicians  of  the  highest  repute  and  the 
validity  of  the  results  obtained  by  them  are  not 
questioned.  There  is  no  doubt  about  the  existence 
of  geometry  of  four  dimensions,  but  it  has  not 
been  proven  that  there  exists  a  corresponding 
space  of  four  dimensions.  Therefore  if  such  a 
thing  as  four-dimensional  space  does  exist,  we 
already  have  at  hand  a  geometry  to  fit  it.  It  is 
important  that  the  reader  should  clearly  perceive 
that  the  concept  of  space  is  one  thing,  while  the 
geometry  of  space  is  quite  another. 

The  geometry  of  four  dimensions  having  then 
a  real  existence  in  the  same  sense  that  geometry 
of  one,  two,  or  three  dimensions  exists,  the  ques- 
tion naturally  arises,  does  there  exist  a  four- 
dimensional  space  which  fits  this  geometry  in  the 
same  sense  that  the  three-dimensional  space  of 


30     The  Fourth  Dimension  and  the  Bible 

our  experience  fits  our  geometry  of  three  dimen- 
sions? This  question  is  one  which  we  cannot 
escape,  it  will  continue  to  haunt  the  thoughtful 
student,  it  is  the  most  interesting  question  con- 
nected with  the  subject.  "The  possibility  that 
we  are  a  part  of  a  four-dimensional  space  with 
physical  limitations  which  confine  us  to  a  three- 
dimensional  space,  and  with  limitations  of  our 
senses  which  prevent  us  from  perceiving  anything 
outside  of  this  space — this  possibility  excites  the 
interest  of  all  who  are  inclined  to  abstract  specu- 
lation. Attempts  may  be  made  to  discover 
physical  proofs  of  such  a  space,  to  build  up 
theories  on  its  basis  that  will  explain  discoveries 
of  modern  physics  as  yet  but  little  understood,  or 
by  it  to  account  for  various  mysterious  phenom- 
ena. Most  of  us  are  satisfied  that  no  real  proofs 
of  the  existence  of  space  of  four  dimensions  will 
be  found  along  these  lines.  Even  a  workable 
hypothesis  based  on  the  (assumed)  existence  of 
four-dimensional  space,  tho  it  might  serve 
temporarily  better  than  any  other  hypothesis, 
would  hardly  justify  a  belief  in  this  existence. 
But  we  do  say  that  the  existence  of  space  of  four 
dimensions  can  never  be  disproved  by  showing 
that  it  is  absurd  or  inconsistent;  for  such  is  not 
the  case.  Nor,  on  the  other  hand,  will  the  most 
elaborate  development  of  analogies  of  different 


On  the  Existence  of  Higher  Spaces       31 

kinds  ever  prove  that  It  does  exist."  (Manning.) 
Kant  not  only  recognized  the  possibility  of  the 
existence  of  spaces  of  more  than  three  dimensions, 
but  he  inferred  their  very  probable  real  existence. 
"If  It  Is  possible,"  he  says,  "that  there  are  de- 
velopments of  other  dimensions  of  space,  It  Is 
also  very  probable  that  God  has  somewhere  pro- 
duced them.  For  his  works  have  all  the  grandeur 
and  variety  that  can  possibly  be  comprised."  Or, 
as  a  mathematician  might  put  It,  why  should  God 
have  created  geometry  of,  say,  four  dimensions, 
If  there  exists  nowhere  a  space  that  will  fit  it? 
God  does  nothing  that  Is  futile  or  unnecessary. 

The  following  quotation  from  "Creation  Ex 
NIhilo"  (Gruber)  will  aid  the  reader  in  maintain- 
ing the  proper  attitude  of  mind  towards  what  has 
gone  before  as  well  as  prepare  him  for  what 
shall  follow. 

"Who  can  limit  the  possible  existence,  within 
and  beyond  the  physical  universe,  even  of  other 
beings  altogether  inconceivable  by  us?  Such 
transcendent  entities,  as  also  God  and  the  human 
soul,  because  not  governed  by  physical  laws,  could 
In  no  way  affect  our  physical  sensorlum.  Not  be- 
ing limited  by  time  and  (our)  space  relations, 
they  might  co-exist  with,  and  around,  us  and 
thruout,  as  well  as  beyond,  the  physical  uni- 
verse— and    this    without    in    the    least    affecting 


32       The  Fourth  Dimension  and  the  Bible 

human  experience  and  consciousness.  ...  As 
experience,  thru  its  proper  avenues  of  ap- 
proach, and  consciousness,  have  their  necessary 
limitations,  as  sources  of  knowledge,  in  their 
limited  points  of  contact  with  self  and  nature,  so 
their  bodyings  forth  in  the  pronouncements  of 
physical  science  do  not  exhaust  reality.  Hence, 
the  great  whole  of  reality  can  never  come  within 
their  compass.  And  this  is  even  true  of  physical 
reality.  But,  as  to  the  transcendent  hyperphysical 
or  spiritual  realities,  consciousness  and  experience, 
unaided — and  therefore  physical  science — would 
forever  have  to  remain  without  a  point  of  contact, 
and  therefore  in  total  ignorance.  No  one  should, 
therefore,  attempt  either  to  set  a  limit  to  existence 
by  his  limited  experience  or  make  his  finite  reason 
the  measure  of  the  immeasurably  complex  uni- 
verse of  the  infinite  and  eternal  God." 

"A  person  born  without  the  sense  of  sight  can- 
not see  light  and  color.  He  cannot  even  form, 
any  real  conception  of  them.  To  him  all  is  dark- 
ness. And,  as  far  as  he  could  by  his  own  powers 
discover,  both  light  and  color  would  have  no  ex- 
istence or  would  belong  to  the  same  category  as 
darkness.  The  man  born  without  the  sense  of 
hearing  can  neither  hear  sound  nor  can  he  even 
form  any  correct  conception  of  sound  and  music. 
And  thus,  to  a  (congenital)  Wind  and  deaf  indi- 


On  the  Existence  of  Higher  Spaces       33 

vidual,  even  this  very  tangible  physical  world  is 
an  entity  altogether  different  from  the  reality. 
Many  of  its  marvelous  phenomena  of  beauty  are 
to  him  totally  non-existent.  And  if,  perchance, 
some  explanation  or  revelation  to  him  of  these 
things  were  attempted  by  a  seeing  and  hearing 
personality,  these  phenomena  would  yet  in  a  sense 
be  utterly  inconceivable  by  him.  He  can  only 
by  touching,  etc.,  acquire  some  indefinite  idea  of 
grosser  forms  and  movements.  But  he  could  not 
perceive  even  any  efect  of  those  subtle  marvelous 
vibrations  that  produce  light  and  color  and 
music." 

"Such  an  individual  lives  in  a  world  of  mar- 
velous beauty,  but  he  beholds  it  not,  nor  can  he 
even  form  any  proper  conception  of  it.  But, 
surely,  it  would  be  an  almost  unpardonable  pre- 
sumption on  his  part  to  deny  the  existence  of  the 
glorious  rainbow  in  the  heavens  and  to  argue  with 
an  entranced  auditor  against  the  existence  of  the 
majestic  symphonies  of  a  Beethoven.  For  these 
things  that  lie  beyond  his  limited  physical  senses, 
he  must  needs  accept  the  testimony  of  those  who 
have  the  necessary  senses  of  perception  to  know 
their  reality." 

"Like  the  blind  and  deaf  individual,  we  stand 
amid  the  wonders  of  nature.  Tho  we  can 
perceive,  in  light  and  color  and  music,  a  minute 


34     The  Fourth  Dimension  and  the  Bible 

fraction  of  the  efect  produced  by  waves  of  ether 
and  waves  of  air,  yet  these  physical  waves  them- 
selves lie  totally  beyond  even  our  natural  sense- 
organs.  Like  that  blind  and  deaf  individual,  who 
thru  his  sense  of  touch,  etc.,  can  form  some  idea 
of  his  physical  environment,  so  we  with  our 
limited  sense-organs  can  acquire  some  knowledge 
of  the  surrounding  universe.  But,  as  in  his  case, 
our  limited  senses  permit  of  but  a  very  partial 
knowledge,  and  beyond  their  range  there  are 
realities  even  in  physical  nature  concerning  which 
we  can  only  speculate  and  which  we  may  never 
know.  We  are  like  children  watching  a  game, 
from  a  point  some  steps  away,  thru  a  small  crack 
or  knot  hole  in  a  boarded  enclosure.  All  appears 
fragmentary  and  partial.  Immeasurably  the 
greater  part  of  the  universe,  from  the  infinitesi- 
mally  small  to  the  universal  whole,  in  ten  thou- 
sand marvels,  lies  beyond  the  range  of  the  whole 
outfit  of  our  physical  senses,  or  is  a  physically 
intangible  reality.  Thus,  the  visible  light  or 
color  spectrum  constitutes  the  record  of  but  a 
minute  fraction  of  the  whole  range  of  the  mys- 
teriously wonderful  perpetual  dance  of  the 
imponderable  ether,  it  constitutes  but  one  of  the 
octaves  of  the  many-octaved  key-board  of  vibra- 
tions. And  yet,  the  other  octaves  no  less  truly 
exist  as  with  our  own  invented  tools  of  investiga- 


On  the  Existence  of  Higher  Spaces       35 

tion  we  are  more  and  more  discovering  from  their 
effects;  but  these  octaves  themselves  we  cannot 
perceive.  If  our  eyes  could  be  so  adjusted  as  to 
enable  us  to  behold  the  whole  range  of  this  funda- 
mental reality,  vistas  of  visions  hitherto  incon- 
ceivable would  lie  before  our  astonished  gaze. 
.  .  .  And  so  might  we  conceive  of  superadded 
transcendent  spiritual  faculties,  if  not  confined 
within  our  physical  organism.  And  thus  the 
transcendent  glories  of  spiritual  realities,  of  an- 
gelic beings,  and  even  of  God,  might  entrance  the 
astonished  perceiving  personality." 

In  what  follows  in  this  text,  higher  dimensional 
spaces,  and  in  particular  the  fourth,  will  often  be 
referred  to  as  if  they  really  existed.  The  reader 
should  clearly  understand  that  this  is  done  only 
to  save  words  by  avoiding  a  needless  repetition 
of  the  statement  that  their  existence  is  only  as- 
sumed. 


CHAPTER  IV 

ABOUT  THE  SUPPOSED  INHABITANTS 
OF  OTHER  SPACES  THAN  OUR  OWN 

As  we  are  now  going  to  make  several  excur- 
sions into  the  realm  of  pure  thought  and  abstract 
speculation  it  will  be  necessary  for  the  reader  to 
approach  the  subject  with  an  open  and  receptive 
mind.  He  will  need  to  occasionally  cut  loose  from 
the  sense  perception  moorings  which  anchor  him 
to  our  material  world,  he  will  often  have  to 
leave  experience  behind  and  reason  wholly  from 
analogy.  A  truly  sympathetic  attitude  will  add 
much  to  the  benefits  that  may  be  derived  from 
the  discussions  which  follow. 

Suppose  that  one,  two,  and  four-dimensional 
spaces  do  exist,  in  the  same  sense  that  the  three- 
dimensional  space  of  our  experience  exists,  and 
let  us  imagine  that  in  each  of  these  spaces  beings 
exist,  each  individual  being  restricted  to  his  own 
particular  space  in  his  movements,  his  percep- 
tions, and  his  understanding,  just  as  we  are  re- 
stricted to  a  three-dimensional  space.  Then  let 
us  by  analogy  try  to  conceive  what  the  attitude 

36 


Supposed  Inhabitants  of  Other  Spaces         2>1 

of  a  four-dimensional  being  living  in  a  four- 
dimensional  space  would  be  towards  us  living  in 
space  of  three  dimensions  by  considering  what 
our  attitude  would  be  towards  beings  living  in 
space  of  one  or  two  dimensions  just  as  if  we  were 
really  conscious  of  their  existence.  For  the  sake 
of  brevity  we  shall  designate  space  of  two 
dimensions  as  Flatland  and  its  inhabitants  as 
Flatlanders,  and  the  space  of  one  dimension  as 
Lineland  and  its  inhabitants  as  Linelanders.  Cor- 
responding to  the  space  in  which  they  live  Line- 
landers  would  have  one  dimension  only  (length) 
and  Flatlanders  two  dimensions  only  (length  and 
breadth). 

FLATLAND 

Let  us  consider  Flatland.  By  imagining  two- 
dimensional  beings  (Flatlanders)  living  in  a  plane 
(Flatland)  and  able  to  glide  about  in  that  plane 
but  absolutely  unable  to  move  out  of  it  or  to 
perceive  anything  of  a  third  dimension,  just  as 
we  are  unable  to  perceive  anything  of  a  fourth 
dimension,  we  may  obtain  a  vivid  idea  of  our 
relation  to  a  possible  four-dimensional  space.  A 
consideration  of  what  would  be  the  attitude  of 
Flatlanders  towards  any  conception  of  space  of 
three  dimensions  makes  clearer  what  should  be 
our  attitude  towards  the  conception  of  a  space  of 


38     The  Fourth  Dimension  and  the  Bible 

four  dimensions.  In  doing  this  we  may  suppose 
that  what  we  call  two-dimensional  matter  is 
really  three-dimensional,  but  with  a  very  slight 
thickness  In  the  third  dimension  which  Flatland- 
ers  are  unable  to  recognize  and  of  which  they  are 
entirely  unconscious.  Thus  we  may  Imagine 
Flatland  In  the  nature  of  a  soap  film  (formed  In 
the  closed  loop  of  a  wire)  In  which  Flatlanders 
live.  Or,  If  we  are  willing  to  accept  the  statement 
of  some  philosophers  and  physicists  that  a  par- 
ticle of  our  matter  has  no  real  substance  (In  the 
sense  In  which  we  ordinarily  use  that  term)  but 
consists  only  of  a  bundle  of  forces,  attractive  and 
repellant,  then  there  Is  no  difficulty  In  thinking  of 
such  forces  as  lying  entirely  in  a  plane  (or  In  a 
line).  All  bodies  In  our  space  are  bounded  by 
surfaces,  in  Flatland  all  bodies  have  one-dimen- 
sional boundaries  (lines  and  curves).  To  the  skin 
(surface)  of  a  human  being  there  would  corre- 
spond the  bounding  contour  (perimeter)  of  a 
Flatlander.  To  other  Flatlanders  he  would  be 
exposed  only  along  his  perimeter,  his  interior 
could  be  reached  by  them  only  thru  an  opening 
in  this  perimeter.  A  Flatlander  surgeon  oper- 
ating on  a  Flatlander  would  have  to  make  an 
incision  in  the  perimeter  of  his  patient  just  as  one 
of  our  surgeons  must  make  an  incision  In  the  skin 
of  a  patient  to  remove  his  appendix.     To  one  of 


Supposed  Inhabitants  of  Other  Spaces         39 

our  surgeons  gifted  with  two-dimensional  vision, 
however,  all  the  interior  organs  of  a  Flatlander 
would  be  exposed  and  he  could  operate  without 
cutting  his  Flatlander  patient  open.  Not  only 
could  we  see  (if  we  had  two-dimensional  vision) 
the  insides  of  Flatlanders  but  we  could  also  see 
the  insides  of  their  houses  and  all  their  closed 
compartments.  To  illustrate  let  us  assume  that 
the  plane  ABCD  on  this  page  is  part  of  Flatland 
and  that  a  Flatlander  inhabiting  it  is  triangular 
shaped  (Fig.  a). 


As  there  is  no  "upward"  or  "downward,"  no 
"above"  or  "below"  in  Flatland,  houses  and 
barns  would  have  no  roofs  or  floors,  only  the  lines 
of  their  sides  (perimeters)  would  be  there. 
Figure  (b)  may  be  taken  to  represent  the  house 
of  a  Flatlander.  He  could  enter  his  house  only 
thru  the  opening  (door)  EF.  There  would  be 
no  difference  between  doors  and  windows  (Fig.  c). 
To  get  to  the  other  side  of  the  line  GH  the 
Flatlander  would  have  to  move  around  one  end 
of  the  line.    The  line  would  form  an  insuperable 


40     The  Fourth  Dimension  and  the  Bible 

barrier  to  him,  he  could  not  cross  it  or  see  over 
or  under  it,  just  as  in  our  space  a  stone  wall  of 
infinite  height  and  depth  would  hinder  our  motion 
and  obstruct  our  line  of  sight.  However,  a  three- 
dimensional  being  (man)  with  two-dimensional 
powers  could  lift  the  Flatlander  up  out  of  the 
plane,  carry  him  across  the  line  and  then  set  him 
down  again  in  the  plane.  To  his  Flatlander 
neighbors  the  Flatlander  during  this  operation 
would  suddenly  vanish  and  remain  invisible  until 
he  reappeared  in  their  world  on  the  other  side 
of  the  line.  This  performance  would  appear  as 
a  miracle  to  the  Flatlanders  because  not  only 
would  it  be  impossible  for  them  to  see  anything 
not  in  their  plane  but  all  their  experiences  being 
confined  to  that  plane  they  could  not  possibly  ex- 
plain the  phenomenon.  In  the  same  manner  the 
Flatlander  might  be  placed  in  or  taken  out  of 
a  closed  room  (Fig.  d)  to  the  utter  mystification 
of  his  friends.  A  robber  from  our  space  with 
two-dimensional  powers  could  therefore  enter 
their  houses  and  steal  their  money  and  jewelry 
from  their  safes  without  breaking  locks  or  doors. 
There  would  be  nothing  in  Flatland  correspond- 
ing to  our  tubes  and  it  would  be  impossible  to 
tie  knots  or  thread  needles  there.  As  a  Flatlander 
could  not  "flip"  a  coin,  to  him  it  would  always  be 
"heads-up"   or   always   "tails-up."      Story  books 


Supposed  Inhabitants  of  Other  Spaces         41 

have  been  written  about  life  in  Flatland,  sup- 
posedly by  Flatland  authors,  in  which  the  whole 
gamut  of  our  human  experiences  are  duplicated 
in  so  far  as  the  restrictions  under  which  they  are 
supposed  to  live  will  allow.  There  are  descrip- 
tions of  life  and  industry  in  Flatland  as  well  as 
tales  of  love  and  war,  and  the  love  stories  close 
with  the  regulation  "they  married  and  lived 
happily  ever  after."  "An  Episode  in  Flatland" 
and  "Scientific  Romances"  by  C.  H.  Hinton; 
"The  Fairyland  of  Geometry"  by  Simon  New- 
comb;  "Flatland"  by  E.  A.  Abbott,  which  has 
also  been  dramatized  and  acted. 

LINELAND 

Let  us  now  consider  Lineland  (one-dimensional 
space)  and  its  supposed  inhabitants,  the  Line- 
landers. 


A '"-"'  I "- 1  lag-— oas— ■^a — — — — fl 

"  D      €     £ 

A  Linelander  (as  C)  would  thruout  life  be  re- 
stricted to  his  line  (of  which  AB  is  a  part)  which 
he  would  conceive  of  as  extending  indefinitely  in 
both  directions.  He  could  slide  freely  back  and 
forth  along  this  line  but  it  would  be  impossible 
for  him  to  move  out  of  it.  He  would  not  be 
conscious  of  nor  could  he  conceive  of  a  second 
dimension;  the  concept  of  a  plane  surface  would 


42     The  Fourth  Dimension  and  the  Bible 

He  entirely  outside  of  his  mental  realm,  just  as 
the  existence  of  our  space  would  be  inconceivable 
to  a  Flatlander.  Bodies  in  Lineland  would  not 
be  bounded  by  surfaces  as  in  our  space  or  by  con- 
tours (perimeters)  as  in  Flatland,  but  by  two 
points,  their  two  extremities.  If  a  Linelander 
encountered  another  similar  being  (as  D)  in  his 
space  neither  could  pass  the  other  for  to  do  so 
would  mean  that  at  least  one  of  them  would  have 
to  move  out  of  the  line,  i.e.,  out  of  their  world. 
Thruout  life  a  Linelander  (as  C)  could  have  only 
two  neighbors  (as  D  and  E),  one  on  each  side 
of  him.  His  experiences  would  be  extremely 
limited,  to  human  beings  life  in  Lineland  would 
be,  to  state  it  mildly,  very  dull  indeed. 

Let  us  now  suppose  that  one  of  these  Line- 
landers  (as  C)  in  some  way  becomes  conscious 
of  the  existence  of  a  second  dimension  and  some- 
how becomes  endowed  with  the  power  to  move 
out  of  the  line  AB  into  a  plane  containing  that 
line  as,  for  instance,  the  plane  of  page  41.  The 
instant  he  moved  out  of  the  line  AB  (his  world) 
he  would  become  invisible  to  his  Lineland  neigh- 
bors D  and  E.  The  vanishing  of  C,  apparently 
into  nowhere,  would  be  utterly  incomprehensible 
to  D  and  E ;  they  could  not  account  for  it  in  any 
way,  it  would  appear  to  them  as  a  supernatural 
phenomenon,  i.e.,  as  a  miracle.     Possessing  the 


Supposed  Inhabitants  of  Other  Spaces         43 

freedom  to  move  about  in  the  plane  (his  new 
world)  the  Linelander  C  could  then  move  back 
into  the  line  AB  (his  old  world).  If  he  should 
return  to  a  point  in  the  line  between  his  old  Line- 
land  neighbors  D  and  E  his  appearance  would  be 
in  the  nature  of  an  apparition  to  both  of  them. 
His  reappearance,  apparently  out  of  nowhere, 
would  be  as  deep  a  mystery  to  them  as  was  his 
disappearance. 

The  happenings  described  in  the  last  paragraph 
may  also  be  imagined  as  occurring  when  the  Line- 
lander  C  has  himself  no  conception  of  a  second 
dimension  and  has  no  power  to  move  out  of  the 
line  AB  (his  world).  We  may  conceive  of  a 
situation  in  which  the  Linelander  C  thru  some 
influence  is  in  the  power  of  a  Flatlander  or  other 
higher-dimensional  being.  This  latter  higher- 
dimensional  being  might  then  take  the  Linelander 
C  out  of  the  line  AB  and  replace  him  at  will,  and 
the  attendant  phenomena  would  appear  the  same 
as  before  to  the  two  Linelanders  D  and  E.  While 
this  was  taking  place  the  Flatlander,  or  other 
higher-dimensional  being  who  was  exercising  his 
power  over  C,  would  be  invisible  to  the  Line- 
landers  D  and  E,  or,  if  not  invisible,  he  would 
appear  to  them  only  as  a  one-dimensional  being 
like  themselves. 

A  Linelander,  therefore,  who  in  some  manner 


44     The  Fourth  Dimension  and  the  Bible 

has  strayed  off  his  line  (one-dimensional  space) 
into  a  plane  (two-dimensional  space)  will  remain 
invisible  to  his  Lineland  neighbors  until  he  re- 
turns to  his  own  world  (the  line). 

Similarly,  a  Flatlander  who  has  moved  out  of 
his  plane  (two-dimensional  space)  into  space  of 
three  dimensions  (our  space)  will  remain  in- 
visible to  his  Flatland  neighbors  until  he  returns 
to  his  own  world  (the  plane). 

Continuing,  the  next  step,  as  suggested  by 
analogy,  would  then  be  the  following.  A  man 
(three-dimensional  being)  who  has  been  trans- 
lated from  our  space  into  a  higher-dimensional 
space  will  remain  invisible  to  earthy  beings  until 
he  returns  again  to  our  space.  We  have  Bible 
records  describing  the  bodily  disappearance  of 
human  beings  from  our  space  in  a  manner 
analogous  to  the  above.  One  case  is  that  of 
Enoch  who  did  not  see  death  but  was  taken  bodily 
out  of  our  world  by  God.  Gen.  5  :24,  Heb.  1 1  :^. 
Another  example  is  the  bodily  translation  of 
Elijah  into  heaven.  II  Kings  2:11.  This  again 
suggests  that  space  of  four  dimensions  may  be 
our  heaven.  Or,  since  the  plural  form  of  the 
word  heaven  occurs  very  often  in  the  Bible,  it  may 
be  that  "the  heavens"  include  not  only  space  of 
four  dimensions  but  also  all  the  other  higher- 
dimensional  spaces.     We  have  no  record  of  a 


Supposed  Inhabitants  of  Other  Spaces         45 

reappearance  in  our  space  of  Enoch,  but  Elijah 
(and  Moses)  did  appear  to  the  disciples  at  the 
transfiguration  of  Christ.  Math.  17  :3,  Mark  9  :4, 
Luke  9:30.  That  neither  Enoch  nor  Elijah  did 
possess  the  inherent  power  to  translate  themselves 
from  earth  to  a  higher  realm  is  evident  from  the 
Bible  statements  that  they  were  taken  by  God 
(the  highest-dimensional  being).  On  two  differ- 
ent occasions  before  his  crucifixion  Christ  passed 
unseen  thru  threatening  multitudes,  since  his  time 
had  not  yet  come.  John  8:59,  10:39.  Did  he 
make  himself  invisible  each  time  by  entering  a 
higher  space?  Was  such  a  space  Christ's  abode 
in  the  interval  between  his  resurrection  and 
ascension  and  during  which  time  he  was  seen  in 
bodily  form  by  his  followers  on  ten  recorded 
occasions?  Each  time  both  his  appearance  and 
his  disappearance  were  miraculous,  each  one  par- 
took of  the  nature  of  a  visit  from  a  higher  to  a 
lower  realm,  from  a  higher  to  a  lower  order  of 
space. 

The  Bible  abounds  with  numerous  accounts  of 
celestial  visitors  (angels,  archangels,  prophets, 
and  God  himself)  revealing  themselves  to  in- 
habitants of  our  earth.  In  view  of  our  discussion 
it  is  near  at  hand  for  us  to  think  of  them  as 
coming  down,  from  the  higher  dimensional  spaces 


46     The  Fourth  Dimension  and  the  Bible 

in  which  they  dwell  and,  after  having  fulfilled 
their  missions  in  our  three-dimensional  space,  to 
again  ascend  on  high.  Did  Paul  have  a  higher- 
dimensional  space  in  mind  when  he  wrote,  "I 
knew  a  man,  whether  in  the  body,  or  out  of  the 
body,  I  cannot  tell,  how  that  he  was  caught  up 
into  paradise,  and  heard  unspeakable  words." 
II  Cor.  12:2,  3.  Read  again  the  Revelations  of 
St.  John,  read  them  with  the  added  light  of  the 
mathematical  hypothesis  of  higher  spaces.  It  will 
be  found  that  much  of  the  fog  with  which  our 
limited  space  perceptions  has  enveloped  it  has 
vanished  and  many  passages  will  take  on  a  new 
meaning;  the  eyes  of  the  soul  will  penetrate 
deeper  than  ever  before  into  the  dim  and  distant 
vistas  which  veil  for  us  the  present  as  well  as 
the  great  unknown  hereafter. 


CHAPTER   V 

MOTION  OF  A  POINT  PERPENDICULAR 
TO  EACH  SPACE 

In  Chapter  II  the  concept  of  the  motion  of  a 
point  perpendicular  to  a  unit  in  each  space  was 
partially  developed.  We  will  now  lead  up  to  an 
extension  of  this  idea  by  making  use  of  the  two 
following  obvious  propositions. 


(a)  A  point  (as  A)  starting  from  the  center 
of  a  line  (as  BC)  and  moving  of  in  a  direction 
perpendicular  to  the  line  will  never  approach  any 
portion  of  the  line  and  will  move  away  at  the  same 
rate  from  its  two  end  points  B  and  C. 


(b)   A  point  (as  A)  starting  from  the  center 
of  a  circle  and  moving  of  in  a  direction  perpen- 

47 


48      The  Fourth  Dimension  and  the  Bible 

dicular  to  its  plane  (as  BC)  will  never  approach 
any  portion   of   the   circumference   of  the   circle 
hut  will  move  away  at  the  same  rate  from  all 
points  on  this  circumference. 
Advancing  another  step  we  get: 


(c)  A  point  (as  A)  starting  from  the  center 
of  a  sphere  and  moving  of  in  a  direction  perpen- 
dicular to  our  three-dimensional  space  will  never 
approach  any  portion  of  the  surface  of  the  sphere 
but  will  move  away  at  the  same  rate  from  all 
points  on  this  surface. 

This  proposition  in  the  Geometry  of  Four 
Dimensions  was  first  proven  by  the  great  astrono- 
mer and  mathematician  Simon  Newcomb,  and  it 
reveals  some  of  the  things,  extraordinary  to  us, 
which  may  be  done  in  four-dimensional  space  if 
such  a  space  exists.  We  may  also  state  the  propo- 
sition as  follows.  Just  as  in  Flatland  a  point  can 
move  In  a  direction  perpendicular  to  a  line,  a 
motion  Inconceivable  to  LInelanders,  and  in  our 
space  a  point  can  move  in  a  direction  perpendicu- 
lar to  two  mutually  perpendicular  lines,  a  motion 


Motion  of  a  Point  Perpendicular        49 

Inconceivable  to  Flatlanders,  so  In  space  of  four 
dimensions  a  point  can  move  in  a  direction  per- 
pendicular to  three  mutually  perpendicular  lines, 
a  motion  which  is  inconceivable  to  us.  Assuming 
that  the  above  sphere  Is  replaced  by  a  material 
hollow  sphere  (as  a  hollow  sealed  glass  globe) 
and  the  point  A  Is  replaced  by  a  grain  of  wheat, 
the  above  proposition  asserts  that  in  space  of 
four  dimensions  the  grain  of  wheat  may  be  placed 
inside  of  the  globe,  or  transferred  from  the  inside 
to  the  outside  of  the  globe,  without  breaking  the 
globe  or  interfering  with  its  structure  in  any 
manner.  This  would  mean  that  in  space  of  four 
dimensions  any  of  our  material  objects  can  be 
taken  out  of,  or  placed  inside  of,  one  of  our 
closed  rooms  or  boxes  without  penetrating  the 
walls  (sides).  A  four-dimensional  robber,  there- 
fore, could  enter  our  houses  without  hindrance 
and  steal  the  money  and  jewelry  in  our  safes  with- 
out breaking  locks  or  doors,  and  none  of  our  jails 
could  hold  him  If  caught.  This  may  be  where 
the  money  we  miss  has  gone,  and  perhaps  the 
articles  we  so  often  lose  have  accidentally  rolled 
into  four-dimensional  space  !  A  four-dimensional 
man  could  eat  one  of  our  soft  boiled  eggs  for 
breakfast  without  breaking  the  shell  and  he  could 
drink  from  one  of  our  bottles  without  removing 
the  cork.    He  could  extract  the  pulp  and  nut  from 


50     The  Fourth  Dimension  and  the  Bible 

one  of  our  plums  without  breaking  the  skin  of 
the  plum ;  in  fact,  he  could  remove  the  kernel  from 
the  nut  without  breaking  the  skin  of  the  plum  or 
the  shell  of  the  nut.  He  could  take  off  his  stock- 
ings without  removing  his  shoes,  and  he  could 
take  out  the  inner  tube  of  one  of  our  automobile 
tires,  repair  it  and  then  replace  it,  without  re- 
moving the  outer  casing.  Just  as  one  of  our  sur- 
geons could  operate  on  a  Flatlander  without 
making  an  incision  in  his  perimeter  so  a  four- 
dimensional  surgeon  could  operate  on  one  of  us 
without  cutting  us  open.  It  is  quite  certain  that 
if  such  a  surgeon  should  locate  among  us  he  would 
very  soon  have  a  large  and  lucrative  practical 
All  of  our  space,  including  the  interior  of  the 
densest  solid,  is  open  to  inspection  and  manipula- 
tion from  the  fourth  dimension. 


— C5 — 

Fig.  a 

An  ordinary  knot  (Fig.  a)  cannot  be  untied  in 
our  space  if  the  two  ends  of  the  cord  are  fixed  (as 
at  A  and  B).  In  four-dimensional  space,  how- 
ever, this  can  be  done  without  unfastening  the 
ends,  in  fact,  a  knot  could  be  tied  or  untied  on  an 
endless  string.     Our  knots  would  therefore  be 


Motion  of  a  Point  Perpendicular        51 

useless  in  four-dimensional  space.  Evidently  our 
chains  would  also  be  useless  because  the  links  in 
them  could  be  separated  without  breaking. 
Corresponding  to  this  in  Flatland  would  be  a  cord 
fastened  at  A  and  B  (Fig.  a)  and  making  a  loop 
around,  say,  C.  This  loop  could  not  be  unwound 
by  a  Flatlander,  but  one  of  us  could  do  it  by 
simply  lifting  that  part  of  the  cord  composing  the 


F/g.  a  Fig.  b 

loop  into  the  third  dimension,  stretching  it  out, 
and  then  replacing  it  in  the  plane   (Fig.  b). 

As  has  already  been  stated,  it  has  been  mathe- 
matically proven,  beyond  the  shadow  of  a  doubt, 
that  if  there  exists  a  space  of  four  dimensions 
then  in  that  space  material  objects  of  our  space 
may  be  placed  in  or  taken  out  of  any  of  our 
hermetically  sealed  closed  compartments  without 
hindrance.  It  has  never  been  satisfactorily 
proven,  however,  that  this  can  be  done  by  a 
human  being;  our  nearest  approach  to  it  is  to 
think  of  it  being  done.    We  can  think  of  the  kernel 


52     The  Fourth  Dimension  and  the  Bible 

of  a  nut  being  extracted  without  breaking  the  shell 
of  the  nut,  in  spite  of  the  fact  that  we  do  not 
understand  how  it  can  be  done.  We  can  think  of 
ourselves  as  entering  or  leaving  closed  compart- 
ments; their  material  walls  do  not  in  the  least 
hinder  our  imagining  ourselves  doing  it.  In  the 
Bible,  however,  we  have  records  telling  of  two 
distinct  occasions  when  Christ  in  the  body  did 
appear  before  his  disciples  assembled  behind 
closed  doors  for  fear  of  the  Jews.  John  20: 
19-23,  26-29.  Oi^  the  first  appearance  Thomas 
was  absent  but  on  the  second  he  was  present  and 
we  read  "then  saith  he  to  Thomas,  reach  hither 
thy  finger,  and  behold  my  hands;  and  reach  hither 
thy  hand  and  thrust  it  into  my  side;  and  be  not 
faithless  but  believing.  And  Thomas  said  unto 
Him,  my  Lord  and  my  God."  That  Christ  ap- 
peared to  his  disciples  on  this  remarkable  occasion 
clothed  in  his  material  body  is  further  emphasized 
in  Luke  24:  36-43.  "Behold  my  hands  and  my 
feet  that  it  is  I  myself :  handle  me,  and  see ;  for  a 
spirit  hath  not  flesh  and  bones,  as  ye  see  me  have. 
.  .  .  And  they  gave  Him  a  piece  of  a  broiled  fish, 
and  of  an  honeycomb.  And  he  took  it,  and  did 
eat  before  them."  Christ,  considered  as  a  higher- 
dimensional  being,  certainly  had  the  power  to  ap- 
pear in  his  body  as  described  above,  or  to  do 
anything  else  which  cannot  be  done  by  us  in  our 


Motion  of  a  Point  Perpendicular         53 

space  of  three  dimensions  but  which  is  possible 
in  our  hypothetical  space  of  four  dimensions  by 
those  who  may  dwell  there.  The  following  re- 
marks on  John  20:19  are  quoted  from  a  Pulpit 
Commentary  on  St.  John.  "It  is  more  than  pos- 
sible— nay,  it  is  entirely  presumable — that  the 
spiritual  body  becomes  possessed  of  additional 
senses,  of  which  we  have  no  conception  or  experi- 
ence; and,  therefore,  the  spirit  clothed  with  such 
body  Is  alive  to  properties  of  matter  and  dimen- 
sions of  space  and  active  forces  all  of  which 
would  be  supernatural  to  us.  Our  Lord  before 
his  passion  gave  numerous  proofs  of  the  domi- 
nance of  his  spirit  over  the  body;  his  repeated 
escapes  from  his  enemies,  his  transfiguration 
glory,  his  superiority  to  gravitation,  in  walking 
upon  the  sea  and  hushing  its  storms.  So  that  he, 
on  this  occasion.  Is  revealing  to  the  world  some 
of  the  functions  of  spiritual  corporeity.  He  is 
manifesting  the  kind  of  life  which  will  eventually 
be  the  condition  of  all  the  redeemed." 


CHAPTER   VI 

ROTATION  OF  SYMMETRICAL 
CONFIGURATIONS 

Let  ABC  and  A'B'C  be  two  lines  In  Llneland 
LL',  the  points  A,B?C,  and  A',B',C'  respectively 

i .  ,^  _____„» L' 

A  be'     ~7.'  1'         c 

being  arranged  in  the  same  order  and  so  located 
that  AB  =  A'B^  AC  =  A'C,  BC  -=  B'C.  One 
line  may  then  be  made  to  coincide  with  the  other 
so  that  A  coincides  with  A',  B  with  B'  and  C  with 
C  by  simply  sliding  one  of  them  along  LL'.  This 
could  be  done  by  a  Linelander. 


\ 
\ 


L' 


h  B     K        0        C    B'  A' 

Let  us  now  consider  the  two  lines  ABC  and 
CB'A'    where,    as    before,    AB  =  A'B',    AC  = 

54 


Rotation  of  Symmetrical  Configurations       55 

A'C,  BC  =  B'C,  but  in  which  the  correspond- 
ing points  are  arranged  in  the  reverse  order.  In 
this  case  the  one  line  cannot,  by  sliding  it  along 
LL",  be  made  to  coincide  with  the  other  so  that  A 
shall  coincide  with  A',  B  with  B',  and  C  with  C, 
and  a  Linelander  could  not  conceive  of  any  other 
way  in  which  it  might  be  done.  A  Flatlander, 
however,  or  one  of  us,  would  at  once  observe  that 
the  corresponding  points  could  be  brought  into 
coincidence  by  rotating  one  of  the  lines  about  O  in 
the  plane  of  the  page,  O  being  the  center  of  CC. 
The  points  A,  B,  C  and  A',  B',  C  respectively  are 
said  to  be  symmetrical  with  respect  to  the  point 
O.  Roughly  stated,  the  line  ABC  may  be  taken 
up  into  Flatland,  turned  over,  and  put  down  again 
on  the  line  C'B'A''  so  that  the  point  A  coincides 
with  A^  B  with  B'  and  C  with  C\  While  this 
rotation  was  taking  place  the  moving  line,  as  in 
the  position  A'^B^'C",  would  be  invisible  to  the 
Linelanders  living  in  LL'  because  it  would  be 
outside  of  their  world;  the  whole  process  would 
appear  to  them  as  a  miracle. 


56     The  Fourth  Dimension  and  the  Bible 


Let  ABC  and  A'B'C  be  two  triangles  in  the 
plane  of  this  page,  such  that  AB  =  A'B',  BC  = 
B'C^  CA  =  C'A',  the  corresponding  sides  of  the 
two  triangles  being  arranged  in  the  same  order. 
One  triangle  may  then  be  made  to  coincide  with 
the  other  by  simply  sliding  it  along  in  the  plane  of 
the  page  until  A  coincides  with  A',  B  with  B', 
and  C  with  C\  This  could  be  done  by  a  Flat- 
lander. 

Y 
Cj 


Let  us  now  consider  the  two  triangles  ABC  and 
A'B'C^  in  which  as  before  AB  =  A'B',  BC  = 
B'C,  and  CA  =  C'A',  but  with  their  correspond- 
ing sides  arranged  in  the  reverse  order.  In  this 
case  the  one  triangle  cannot,  by  sliding  it  along 
in  the  plane,  be  made  to  coincide  with  the  other, 
and  a  Flatlander  could  not  conceive  of  any  other 
way  in  which  it  might  be  done.  The  reader  should 
verify  the  first  statement  in  the  last  sentence  by 
cutting  a  triangle  out  of  cardboard  to  fit,  say  ABC, 
and  then,  by  sliding  it  along  the  page,  try  to 
make  it  fit  on  A'B'C^  A  three-dimensional 
mathematician  would  observe,  however,  that  the 


Rotation  of  Symmetrical  Configurations       57 


two  triangles  ABC  and  A'B'C  are  symmetrical 
with  respect  to  the  line  XY  (corresponding  points 
on  the  two  triangles  being  located  on  the  same 
perpendicular  to  XY  and  at  equal  distances  from 
it)  and  that  they  can  be  brought  into  coincidence 
by  rotating  one  (with  the  plane  in  which  it  lies) 
about  the  axis  of  symmetry  XY  thru  three-dimen- 
sional space  until  it  coincides  with  the  other. 
That  is,  one  triangle  must  be  taken  up  into  our 
space,  turned  over,  and  put  down  on  the  other. 
To  Flatlanders  this  operation  would  be  an  un- 
solvable  mystery. 


r/ff.a 


rig.b 


If  one  of  us  should  take  a  Flatlander  (as 
Fig.  a)  and  turn  him  over  thru  our  space  back 
into  his  plane  (as  Fig.  b)  he  would  become  a  sort 
of  mirror  reflection  of  his  former  self.  His  heart 
would  now  be  found  on  his  right  side  instead  of 
his  left,  and  if  he  was  right  handed  before  he 
would  now  be  left  handed.  While  he  was  being 
turned  over  in  our  space  he  would  be  invisible  to 
his  Flatland  neighbors;  first  he  would  apparently 


58      The  Fourth  Dimension  and  the  Bible 

vanish  before  their  eyes  into  nowhere  and  then 
as  suddenly  he  would  reappear  like  an  apparition. 
Before  taking  the  next  step  in  our  discussion 
let  us  digress  a  trifle  to  point  out  that  geometrical 
propositions  relating  to  configurations  in  a  par- 
ticular space  may  sometimes  be  proven  with 
greater  ease  by  operating  in  a  higher  space  than  if 
we  are  restricted  to  the  original  space.  A  case  in 
point  is  the  plane  geometry  proposition  that  two 
triangles  are  equal  if  the  three  sides  of  one  are 
equal  respectively  to  the  three  sides  of  the  other. 
In  practically  all  the  plane  geometry  text-books 
used  in  our  schools  the  proof  of  this  proposition 
requires  a  rotation  of  one  of  these  triangles  about 
one  of  its  sides.  But  this  means  that  the  pupil  is 
abandoning  the  space  of  his  plane  geometry  to 
operate  in  the  space  of  solid  geometry  (three 
dimensions).  In  like  manner  there  are  many 
theorems  in  both  plane  and  solid  geometry  which 
can  be  proven  with  greater  ease  by  using  geometry 
of  four  dimensions  than  by  using  the  methods 
found  in  our  school  books.  In  fact  both  plane 
and  solid  geometry  have  been  greatly  enriched  by 
the  discovery  thru  the  geometry  of  four  dimen- 
sions of  a  number  of  new  theorems.  Just  as  solid 
geometry  throws  a  new  light  on  plane  geometry 
and  enables  us  to  solve  problems  in  the  plane 
with  greater  ease  than  if  we  confined  ourselves 


Rotation  of  Symmetrical  Configurations       59 

to  the  plane,  so  it  has  been  found  that  geometry 
of  four  dimensions  illuminates  both  plane  and 
solid  geometry,  and  has  made  easy  or  rendered 
possible  the  solution  of  many  problems  both  in 
the  plane  and  in  our  three-dimensional  space 
which,  without  the  concepts  of  geometry  of  four 
dimensions,  would  have  been  very  difficult  of  solu- 
tion to  say  the  least.  Schubert  has  illustrated  this 
by  means  of  some  very  interesting  examples. 
We  now  return  to  our  line  of  discussion. 


Let  ABCD  and  A'B'CD'  be  two  tetrahedrons 
in  our  space  having  all  the  faces  of  one  equal  to 
the  corresponding  faces  of  the  other  and  arranged 
in  the  same  order.  By  superimposing  one  on  the 
other  we  can  make  one  coincide  with  the  other 
without  difficulty. 


/I 


1/ 


6o     The  Fourth  Dimension  and  the  Bible 

Let   us    now    consider   the    two    tetrahedrons 
ABCD  and  A'B'C^D'  in  which,  as  before,  all  the 
faces  of  one  are  equal  to  the  corresponding  faces 
of  the  other,  but  arranged  in  the  reverse  order. 
No  amount  of  manipulation  will  now  enable  us 
to  bring  the  two  tetrahedrons  into  coincidence, 
and  no  human  being  can  conceive  of  any  way  in 
which  it  can  be  done.     This  task  is  as  impossible 
to  us  as  was  the  task  of  the  Linelander  to  bring 
the  symmetrical  lines  ABC  and  CB'A'  into  coin- 
cidence (p.  54)  or  the  task  of  the  Flatlander  to 
make   the   two   symmetrical  triangles  ABC   and 
A''B''C''  coincide  (p.  ^6).    Mathematicians  of  the 
highest  standing,  however,  have  proven  that  it 
can  be  done  in  four  dimensional  space  by  rotating 
one  of  the  tetrahedrons  about  their  plane  of  sym- 
metry XY  (corresponding  points  of  the  two  tetra- 
hedrons being  located  on  the  same  perpendicular 
to  XY  and  at  equal  distances  from  it),  a  kind  of 
rotation  that  is  impossible  in  our  space  of  three 
dimensions.     Or,  stating  it  roughly,  one  of  the 
tetrahedrons  must  be  taken  up  into  space  of  four 
dimensions,  turned  over,  and  then  brought  back 
into   concidence   with   the   other.      One   of  these 
tetrahedrons  corresponds  to  the  mirror  image  of 
the  other,  the  mirror  being  in  the  plane  of  XY. 
While  this  rotation  is  taking  place  the  moving 


Rotation  of  Symmetrical  Configurations       6i 

tetrahedron  would  be  invisible  to  us;  just  as  the 
rotating  line  was  invisible  to  the  Linelander  (p. 
54)  and  the  rotating  triangle  was  invisible  to  the 
Flatlander  (p.  56). 

When  you  look  at  your  image  in  a  mirror,  the 
right  and  left  sides  of  your  body  appear  to  have 
exchanged  places.  A  mole  on  your  right  cheek 
is  seen  on  the  left  cheek  of  your  image.  Your 
image  is  symmetrical  to  your  body  with  respect 
to  the  plane  of  the  mirror.  You  may  go  behind 
the  mirror  to  the  place  where  your  image  ap- 
peared but  you  cannot  take  the  exact  position  of 
the  image,  or  rather,  you  cannot  make  your  body 
coincide  in  every  detail  with  that  position.  Turn 
and  twist  about  as  much  as  you  please  In  our 
space  you  cannot  make  your  right  and  left  sides 
exchange  places.  But  a  four-dimensional  being 
could  do  that  very  thing  to  you  by  rotating  you 
about  a  plane  In  his  space.  And,  what  is  more, 
no  part  of  your  body  would  be  dislocated  during 
the  operation,  you  would  not  experience  any  tear- 
ing or  wrenching  of  the  tissues  of  your  body. 
After  this  "reversal"  or  "twist"  In  space  of  four 
dimensions,  during  which  you  would  be  invisible 
to  all  other  human  beings,  everything  on  you 
would  also  be  changed  from  right  to  left  and 
vice-versa,  even  the  seams  and  buttons  on  your 
clothes,  every  dimple  and  wrinkle  on  your  face 


62     The  Fourth  Dimension  and  the  Bible 

and  every  hair  on  your  head.  Your  point  of  view 
would  be  completely  turned  around,  so  that  every- 
one else  would  appear  to  you  to  be  reversed.  The 
letters  on  this  page  would  seem  reversed  to  you, 
i.e.,  as  they  really  do  appear  on  the  printer's  type 
form.  The  hands  of  a  clock  would  appear  to 
you  as  going  backward  and  the  sun  to  rise  in  the 
west  and  set  in  the  east;  the  whole  world  would 
appear  to  you  as  a  looking-glass  world.  When 
you  look  into  a  mirror  you  see  a  picture  of  the 
actual  "reversal"  or  "twist"  that  can  be  imparted 
to  material  bodies  in  the  space  of  four  dimensions, 
but  which  cannot  be  brought  about  in  our  space 
of  three  dimensions. 

Our  two  hands  appear  to  be  at  least  approxi- 
mately of  the  same  size,  and  shape  but  we  can 
never  manipulate  one  so  that  it  shall  occupy  the 
same  space  that  has  been  occupied  by  the  other; 
we  cannot  put  a  right  glove  on  the  left  hand,  or 
vice-versa.  The  reason  for  this  is  that  our  two 
hands  are  symmetrical  with  respect  to  some  plane 
in  our  space  and,  like  the  two  symmetrical  tetra- 
hedrons shown  on  p.  59,  they  cannot  be  made  to 
coincide  by  any  operation  entirely  confined  to  our 
space  of  three  dimensions.  For  the  same  reason 
we  cannot  wear  a  left  shoe  on  the  right  foot,  or 
vice-versa.  A  four-dimensional  man,  however, 
can  change  a  right  shoe  into  a  left  shoe  by  rotat- 


Rotation  of  Symmetrical  Configurations       63 

ing  It  about  a  plane  In  his  space.  If  a  one  armed 
man  in  our  space  should  In  some  way  become 
endowed  with  the  power  to  rotate  bodies  about 
planes  In  four-dlmenslonal  space  he  could  use  both 
of  an  ordinary  pair  of  gloves  because  he  could 
change  either  one  Into  the  other,  so  that  he  would 
have  either  two  rights  or  two  lefts. 

Instead  of  rotating  material  bodies  about  a 
plane  In  space  of  four  dimensions  as  described 
above  It  has  been  proven  that  a  curved  surface 
may  also  be  used  as  the  "axis"  of  rotation  by 
allowing  for  a  slight  degree  of  distortion.  This 
leads  to  very  interesting  results  If  the  rotation  is 
applied  to  flexible  material  surfaces  like  a  curved 
sheet  of  rubber.  The  two  surfaces  of  the  sheet 
of  rubber  may  be  made  to  exchange  places  by 
rotating  them  about  the  sheet  itself  in  space  of 
four  dimensions.  During  the  rotation  there 
would  be  no  Interference  betw^een  portions  of  the 
surfaces  so  that  it  may  be  done  with  surfaces  that 
are  closed  (like  a  sealed  hollow  rubber  ball)  as 
well  as  with  surfaces  that  are  open  (like  a  sheet 
of  rubber). 

This  means  that  a  hollow  rubber  ball  (Fig.  a) 
could  be  turned  Inside  out  without  tearing  just  as 
we  may  turn  a  rubber  band  or  the  section  of  an 
inner  tire  tube  (Fig.  b)  Inside  out.     In  both  cases 


64     The  Fourth  Dimension  and  the  Bible 


the  inside  and  outside  surfaces  would  exchange 
places. 


Fig  a 


rig.b 


Not  only  in  the  animal  kingdom  do  we  find  this 
wonderful  symmetry  (right  and  left  handedness) 
exemplified  in  the  bodies  of  all  creatures,  as  well 
as  in  most  of  the  interior  organs  of  those  bodies, 
but  it  also  appears  very  often  in  the  vegetable 
kingdom  as  shown  by  the  symmetry  of  the  two 
sides  of  a  leaf,  and  in  flowers,  bulbs,  nuts,  fruits, 
etc.  In  fact,  the  right  and  left  rotation  (twist) 
which  is  characteristically  four-dimensional  is  of 
very  frequent  occurrence  in  nature.  A  beam  of 
polarized  light  (whose  wave-vibrations  are  all  in 
one  plane)  is  rotated  either  to  the  right  or  to  the 
left  on  passing  thru  certain  organic  substances 
(sugars,  starches)  just  as  we  might  hold  a  ribbon 
by  the  ends  and  give  it  a  twist  to  the  right  or  left. 
Two  forms  of  sugar  found  in  honey  called  dex- 
trose and  levulose  owe  their  names  to  the  fact 
that  one  rotates  a  polarized  beam  to  the  right 
(dextra,  "right  hand")  and  the  other  to  the  left 
(laeva,   "left  hand").     In   chemical  constitution 


Rotation  of  Symmetrical  Configurations       65 

they  are  exactly  the  same,  they  are  both  repre- 
sented by  the  same  formula,  and  yet  they  rotate 
the  plane  of  polarization  of  a  beam  of  light  in 
opposite  directions.  Such  substances  are  called 
"isomeric."  A  molecule  of  each  of  the  two  sub- 
stances contains  the  same  number  of  each  kind 
of  atom,  the  difference  in  the  effect  on  light  being 
accounted  for  by  a  different  arrangement  or  group- 
ing of  the  atoms  in  the  two  molecules. 

The  following  discussion  will  explain  what  rela- 
tion this  has  to  a  possible  four-dimensional  space. 

In  a  plane  three  points  (as  the  vertices  of  the 


r/^.  a  F'ig.  b 


triangle  In  Fig.  a),  and  no  more,  can  be  located 
so  that  the  three  distances  between  them  shall 
be  independent  of  each  other,  i.e.,  any  one  of  the 
distances  may  change  without  changing  the  others. 

In  our  space  of  three  dimensions  four  points 
(as  the  vertices  of  the  tetrahedron  in  Fig.  b),  and 
no  more,  can  be  located  so  that  the  six  distances 
between  them  shall  be  independent. 

In  space  of  four  dimensions  five  points,  and 


66     The  Fourth  Dimension  and  the  Bible 

no  more,  can  be  located  so  that  the  ten  distances 
between  them  shall  be  independent,  something  that 
cannot  be  done  in  our  space. 

If  then  a  molecule  consists  of  five  atoms  we 
cannot  in  our  space  alter  the  distance  between  two 
of  them  without  at  least  altering  some  second  dis- 
tance. "But  if  we  imagine  the  centers  of  the 
atoms  placed  in  four-dimensional  space,  this  can 
be  done;  all  the  ten  distances  which  may  be  con- 
ceived to  exist  between  the  five  points  will  then 
be  independent  of  one  another.  To  reach  the 
same  result  in  the  case  of  six  atoms  we  must  as- 
sume a  five-dimensional  space;  and  so  on.  Now, 
if  the  independence  of  all  possible  distances  be- 
tween the  atoms  of  a  molecule  is  absolutely  re- 
quired by  theoretical  chemical  research,  the 
science  is  really  compelled,  if  it  deals  with  mole- 
cules of  more  than  four  atoms,  to  make  use  of  the 
idea  of  a  space  of  more  than  three  dimensions. 
This  idea  is,  in  this  case,  simply  an  instrument 
of  research,  just  as  are,  also,  the  ideas  of  mole- 
cules and  atoms — means  designed  to  embrace  in 
a  perspicuous  and  systematic  form  the  phenomena 
of  chemistry  and  to  discover  the  conditions  under 
which  new  phenomena  can  be  evoked.  Whether 
or  not  a  four-dimensioned  space  really  exists  is 
a  question  whose  insolubility  cannot  prevent  re- 
search from  making  use  of  the  idea,  exactly  as 


Rotation  of  Symmetrical  Configurations       67 

chemistry  has  not  been  prevented  from  making 
use  of  the  notion  of  atom,  altho  no  one  really 
knows  whether  the  things  we  call  atoms  exist  or 
not."     (Schubert.) 

For  example,  at  least  eight  different  alcohols 
have  been  discovered  having  the  same  formula 
C5  H12  O.  Chemists  account  for  these  different 
compounds  by  supposing  that  there  is  a  different 
arrangement  or  grouping  of  the  five  carbon,  or 
C-atoms,  in  each  compound,  something  which 
would  be  possible  in  space  of  four  dimensions  but 
not  possible  in  our  space.  In  other  words,  should 
it  become  necessary  for  the  explanation  of  the 
structure  of  this  molecule  to  assume  that  the  five 
carbon  atoms  in  it  are  equidistant  from  each  other 
(the  simplest  possible  case)  we  would  also  have 
to  assume  the  existence  of  four-dimensional  space, 
because  in  that  space,  and  not  in  our  space,  is  it 
possible  to  locate  five  points  equidistant  from  each 
other. 

There  are  two  varieties  of  tartaric  acid  which 
crystaUize  into  symmetrical  forms,  forms  bearing 
the  relation  of  object  to  mirror-image  as  here 
shown. 


68     The  Fourth  Dimension  and  the  Bible 

Investigators  have  found  that  one  of  these 
crystals  apparently  changes  into  the  other  with- 
out chemical  resolution  and  reconstitution  and 
without  any  manifestation  of  force.  If  it  could  be 
proven  without  the  shadow  of  a  doubt  that  this 
is  what  actually  happens,  then  this  would  point 
to  the  existence  of  space  of  four  dimensions,  be- 
cause in  that  space,  and  not  in  ours,  is  it  possible 
for  a  right-handed  shape  to  change  into  its  sym- 
metrical left-handed  shape  by  a  simple  movement. 
Certain  snails,  exactly  alike  in  other  respects, 
differ  similarly  in  that  some  are  coiled  to  the  right 
and  others  to  the  left.  An  astonishing  phenome- 
non connected  with  these  snails  is  the  fact  that 
their  juices  have  the  property  of  rotating  a  polar- 
ized beam  of  light  to  the  right  or  to  the  left 
corresponding  to  the  direction  in  which  the  snails 
themselves  are  coiled.  This  suggests  that  their 
external  form  is  the  expression  of  an  internal 
difference  due  to  a  right  or  left  twist  of  their 
atoms  caused  by  a  four-dimensional  force.  It 
has  been  suggested  that  such  a  four-dimensional 
twist  runs  thru  all  living  forms,  and  that  the  life 
force  is,  in  part  at  least,  four-dimensional. 


CHAPTER   VII 

SECTIONS  OF  BODIES  IN  ONE  SPACE 
MADE  BY  SPACES  OF  A  LOWER 
ORDER.  CONSERVATION  OF  MAT- 
TER AND  ENERGY 

Suppose  we  pass  a  circular  cone  downward  per- 
pendicularly thru  Flatland  XY  (see  figure).  The 
cross  sections  of  the  cone  made  by  the  plane  of 


Flatland  will  be  circles  (as  AB).  The  Flatland- 
ers  living  in  the  plane  would  see  at  first  only  a 
point  as  the  apex  of  the  cone  entered  their  plane, 
then  they  would  see  a  circle  gradually  increasing 
to  the  size  of  the  base  of  the  cone,  when  suddenly 
every  trace  of  the  cone  would  vanish.  The  Flat- 
landers  would  have  no  conception  whatever  of 
the  cone  as  we  know  it,  indeed,  they  would  not 
suspect  that  the  two-dimensional  object   (circle) 

69 


70     The  Fourth  Dimension  and  the  Bible 

which  they  had  seen  had  anything  to  do  with  a 
three-dimensional  solid  at  all,  even  if  they  had  an 
inkling  of  the  existence  of  our  space.  The  phe- 
nomenon would  appear  to  them  as  the  creation  of 
a  new  two-dimensional  body  (circle)  which  con- 
tinued to  grow  larger  and  larger  and  then  sud- 
denly vanished.  The  alpha  and  omega  of  this 
phenomenon  would  be  deep  mysteries  to  them, 
just  as  birth  and  death  are  to  us.  In  like  manner 
It  may  be  that  each  one  of  us  is  in  reality  a  four- 
(or  higher-)  dimensional  being,  our  personality 
(as  we  are  known  to  ourselves  and  others)  being 
the  section  of  our  real,  our  higher  self,  made  by 
the  three-dimensional  space  of  our  experience. 
Like  the  cone  passing  thru  Flatland,  we  arrive  into 
this  three-dimensional  world  from  some  unknown 
shore,  i.e.,  are  born;  we  grow  up  thru  childhood, 
youth,  middle  age,  old  age ;  we  die,  and  our  souls, 
our  real,  our  higher  selves,  pass  into  the  great 
unknown  hereafter.  And  thruout  life  we  have 
been  thought  of  by  ourselves  and  our  fellowmen 
as  three-dimensional  beings  only.  Perhaps  all  liv- 
ing things,  as  they  appear  to  us,  are  the  sections 
of  four-  (or  higher-)  dimensional  creatures  made 
by  our  space.  "Viewing  human  life  from  this 
standpoint,  the  conclusion  may  be  reached  that  I, 
as  I  write  this,  am  merely  that  section  of  my  four- 
dimensional  self  that  happens  to  be  passing  thru 


Sections  of  Bodies  in  One  Space  71 

this  world  at  this  moment,  and  that  the  whole 
of  me,  from  my  birth  to  my  death,  is  a  four- 
dimensional  entity;  that  the  past  and  the  future 
are  past  and  future  only  in  a  three-dimensional 
sense,  and  that  in  a  four-dimensional  sense  the 
past  and  future  are  present — that  is,  both  what 
was  and  will  be,  is."     (Taylor.) 

Suppose  now  that  a  particular  Flatlander  had 
In  some  way  acquired  the  Intellectual  power  to 
sum  up,  in  his  mind,  in  a  moment  of  time,  all  the 
cross  sections  (circles)  of  the  cone  he  had  seen. 
He  would  then  obtain  a  mental  picture  of  the  cone, 
not  as  we  see  it,  but  as  a  body,  not  of  two  dimen- 
sions, but  as  a  body  having  three  dimensions,  time 
being  the  third  dimension.  In  like  manner,  by 
a  concentrated  mental  effort,  we  may  in  a  moment 
of  time  assemble  all  the  impressions  we  have  re- 
ceived of  some  friend  or  relative  with  whom  we 
have  been  intimately  associated  from  his,  or  her, 
birth  to  death.  The  mental  image  thus  obtained 
would  include  all  the  years  of  his  or  her  life,  the 
whole  span  of  life  of  that  individual  would  be 
compressed  into  a  moment  of  time;  and  this 
mental  image  would  have  four  dimensions,  time 
being  the  fourth  dimension.  Mental  vision  has 
therefore  been  called  four-dimensional  by  some 
writers.  All  of  us  have  in  our  dreams  experienced 
this  crowding  together  of  a  long  series  of  occur- 


72      The  Fourth  Dimension  and  the  Bible 

rences  into  an  instant  of  time.  There  are  no  re- 
strictions of  time,  or  restrictions  of  the  space  of 
our  experience,  in  dreamland.  "The  unthinkable 
velocity  of  time  in  dreams  may  be  inferred  from 
the  fact  that  between  the  movement  of  impact  of 
an  impression  at  the  sense  periphery  and  its  re- 
ception at  the  center  of  consciousness — moments 
so  closely  compacted  that  we  think  of  them  as 
simultaneous — a  coherent  series  of  representa- 
tions take  place,  involving  what  seem  to  be  pro- 
tracted periods  for  their  unfoldment.  Every 
reader  will  easily  call  to  mind  dream  experiences 
of  this  character,  in  which  the  long-delayed 
denouement  was  suggested  and  prepared  for  by 
some  extraneous  sense  impression,  showing  that 
the  entire  dream  drama  unfolded  within  the  time 
it  took  that  impression  to  travel  from  the  skin  to 
the  brain."  (Bragdon).  Perhaps  dreamland  is 
located  in  four-dimensional  space. 

A  man  who  has  experienced  the  sensations  of 
a  dying  person  as,  for  instance,  a  man  who  be- 
lieved he  was  drowning,  tells  us  that  in  an  instant 
the  whole  panorama  of  his  past  life  was  spread 
out  before  him  in  a  vivid  momentary  perspective. 
"Memory  is  a  carrying  forward  of  the  past  into 
the  present,  and  the  fact  that  we  can  recall  a  past 
event  without  mentally  rehearsing  all  the  inter- 
mediate happenings  in  Inverse  order,  shows  that 


Sections  of  Bodies  in  One  Space  73 

in  the  time  aspect  of  memory  tliere  is  simultaneity 
as  well  as  sequence."  If  our  mental  vision  is  four- 
dimensional  it  points  to  the  possibility  that  our 
mental  or  spiritual  self  is  four-  (or  higher-) 
dimensional.  If  time  is  but  the  way  in  which  we 
perceive  the  fourth  dimension,  then  our  spiritual 
selves,  being  higher-dimensional,  are  independent 
of  time,  above  time,  outside  of  time,  that  is, 
eternal.  In  this  higher  space  our  grammatical 
tenses  would  then  have  no  meaning.  Perhaps  this 
explains  why  these  tenses  as  used  in  the  Bible 
sometimes  appear  "mixed"  to  us.  Jesus  said  "be- 
fore Abraham  was,  I  am."  "For  a  thousand 
years  in  thy  sight  are  but  as  yesterday  when  it  is 
past."  Ps.  90:4.  That  God,  the  highest-dimen- 
sional being  is  not,  like  we,  restricted  by  time 
limitations  is  clear.  "One  day  is  with  the  Lord  as 
a  thousand  years,  and  a  thousand  years  as  one 
day."  II  Peter  3  :8.  "Behold,  thou  hast  made 
my  days  as  an  handbreadth;  and  mine  age  is  as 
nothing  before  thee."     Ps.  39:5. 

Another  interesting  illustration  similar  to  that 
of  the  cone  passing  thru  Flatland  (p.  69)  is  that 
of  a  wire  bent  in  the  form  of  a  circular  helix 
(coiled  wire  spring)  which  is  passed  thru  Flat- 
land  so  that  its  axis  is  perpendicular  to  the  plane 
of  Flatland  (see  figure).  The  cross  section  of 
the  wire  made  by  the  plane  (as  A)  will  be  a  very 


74     The  Fourth  Dimension  and  the  Bible 


small  ellipse  (two-dimensional).  If  the  wire  is 
passed  downward  the  Flatlanders  would  see  this 
eUipse  travel  in  the  circle  BC  in  one  direction, 
while  if  the  wire  is  passed  upward  it  would  travel 
in  the  opposite  direction.  The  "birth"  of  this 
ellipse  as  the  wire  entered  the  plane,  its  "death" 
when  it  left  the  plane,  and  its  behavior  in  the 
interim — all  would  be  inscrutable  mysteries  to 
the  Flatlanders  living  in  that  plane. 

It  is  evident  that  if  the  cone  to  which  we  re- 
ferred on  p.  69  is  passed  obliq^uely  thru  the  plane, 
then  the  sections  will  no  longer  be  circles  but 
ellipses,  parabolas,  hyperbolas,  or  straight  lines, 
all  depending  on  the  angle  which  the  axis  of  the 
cone  makes  with  the  plane.  To  further  illustrate 
that  the  kind  of  section  made  in  a  given  case  will 
depend  not  only  on  the  shape  of  the  solid  but  also 
on  how  the  solid  enters  the  plane,  let  us  assume 
that  the  same  cube  passes  thru  the  plane  several 
times,  each  time  with  a  new  orientation  with  re- 
spect to  the  plane. 


Sections  of  Bodies  in  One  Space         75 


F/j-  a 


Fig.b 


Fig.  c 


If  a  face  of  the  cube  is  always  parallel  to  the 
plane,  then  all  the  sections  made  will  be  equal 
squares  (Fig.  a). 

If  an  edge  of  the  cube  (and  no  face)  is  always 
parallel  to  the  plane  then  all  the  sections  made  will 
be  rectangles  except  the  first  and  last  which  would 
be  lines,  i.e.,  the  top  and  bottom  edges  (Fig.  b). 

If  no  edge  or  face  is  parallel  to  the  plane,  i.e., 
the  cube  is  passed  thru  obliquely,  the  first  section 
will  be  a  point  (a  corner  of  the  cube),  then  they 
would  be  triangles,  then  polygons,  then  again  tri- 
angles, and  finally  the  last  one  would  again  be 
a  point  (Fig.  c). 

Since  we  live  in  space  of  three  dimensions  the 
above  phenomena  are  fully  understood  by  us,  but 
to  Flatlanders  viewing  the  appearance,  behavior, 
and  disappearance  of  these  sections  In  their  plane, 
it  would  all  be  utterly  Inexplicable  because  they 
have  knowledge  and  experience  only  of  two- 
dimensional  bodies.  Least  of  all  would  they  sus- 
pect that  sections  so  unlike  were  produced  by  the 


76     The  Fourth  Dimension  and  the  Bible 

passing  of  the  same  higher-dimensional  body  thru 
their  world.  If  then,  as  it  has  been  suggested, 
the  personality  of  a  man  is  the  section  of  his  real, 
his  higher-dimensional  self,  made  by  the  space  of 
our  experience,  it  makes  it  clearer  to  us  why  the 
personalities  of  all  the  people  we  know  have  such 
characteristic  differences.  Both  as  to  mind  and 
body  they  all  seem  to  have  been  cast  in  different 
moulds.  All  this  suggests  that  the  personality  of 
a  man  may  depend  not  only  on  the  character  of 
his  higher-dimensional  self  but  that  it  is  also  influ- 
enced by  the  manner  in  which  that  higher  self  was 
projected  into  our  world. 

In  the  preceding  paragraphs  of  this  chapter  the 
reader  naturally  thought  of  the  cone,  helix,  and 
cube  as  substantial  solids,  while  the  sections 
(points,  circles,  triangles,  lines,  etc.)  of  these 
solids  made  by  the  planes  thru  which  they  were 
passing  impressed  him  with  their  changing,  fleet- 
ing, and  evanescent  character.  Passing  now  to  the 
next  higher  order  of  space,  analogy  suggests  the 
above  relations  as  one  explanation  of  the  transi- 
tory character  of  our  lives;  they  may  be  merely 
the  sections  of  our  spiritual,  our  higher-dimen- 
sional selves  made  by  the  three-dimensional  space 
of  our  experience.  Many  passages  in  the  Bible 
emphasize  the  ephemeral  character  of  our  earthly 
existence,  while  others  point  out  the  higher,  the 


Sections  of  Bodies  in  One  Space         77 

divinely  transcendental  origin  and  nature  of  man. 
"As  for  man,  his  days  are  as  grass;  as  a  flower  of 
the  field,  so  he  flourisheth.  For  the  wind  passeth 
over  it,  and  it  is  gone ;  and  the  place  thereof  shall 
know  it  no  more."  Ps.  103:15,  16.  "For  what 
is  your  life?  It  is  even  a  vapour,  that  appeareth 
for  a  little  time  and  then  vanisheth  away."  James 
4:14.  "Man  that  is  born  of  a  woman  is  of  few 
days,  and  full  of  trouble.  He  cometh  forth  like 
a  flower,  and  is  cut  down :  he  fleeth  also  as  a 
shadow,  and  continueth  not."  Job  14:1,  2. 
"Mine  age  is  departed,  and  is  removed  from  me 
like  a  shepherd's  tent."  Is.  38:12.  "So  God 
created  man  in  his  own  image,  in  the  image  of 
God  created  he  Him."  Gen.  i  :27.  "For  thou 
hast  made  him  a  little  lower  than  the  angels,  and 
hast  crowned  him  with  honor  and  glory."  Ps. 
8  :5.  "For  in  him  we  live  and  move  and  have  our 
being."    Acts  17  :28. 

In  this  connection  it  is  of  interest  to  refer  to 
the  method  employed  by  the  biologist  to  deter- 
mine the  interior  structure  and  composition  of 
solids.  For  instance,  by  cutting  an  apple  into  a 
large  number  of  thin  slices  of  uniform  thickness 
and  arranging  them  in  the  proper  sequence  on  a 
flat  surface  a  clear  concept  of  the  structure  of  the 
apple  may  be  obtained.  That  is,  he  mentally  sums 
up  the  total  of  the  impressions  on  his  mind  made 


78      The  Fourth  Dimension  and  the  Bible 

by  these  slices  and  thus  obtains  a  mental  image 
of  the  Interior  as  well  as  the  exterior  of  the  apple. 
Thus,  if  the  apple  happens  to  be  worm-eaten  it  is 
easy  to  trace  the  passage  made  by  the  worm,  as 
well  as  to  locate  the  worm,  by  noting  the  relative 
locations  of  the  series  of  worm  holes  in  the  slices. 
This  method  is  of  the  greatest  utility  to  the 
biologist  in  the  study  of  the  structure  and  compo- 
sition of  the  embryos  or  organs  of  animate  beings. 
For  instance,  the  embryo  of  a  frog  is  cut  into  a 
large  number  of  sections  by  means  of  the  micro- 
tome, an  instrument  constructed  for  this  purpose, 
so  thin  that  they  are  practically  transparent  under 
the  microscope.  These  slices,  which  are  prac- 
tically plane  sections  of  the  embryo,  are  then 
mounted  on  glass  plates  by  means  of  a  perfectly 
transparent  adhesive  and  studied  under  the 
microscope.  These  plates,  called  slides,  when 
consecutively  arranged  in  a  series,  will  vividly  pro- 
tray  the  most  minute  detail  of  the  structure  of  the 
embryo.  By  means  of  these  slides  the  biologist 
Is  able  to  mentally  reassemble  all  these  sections  In 
their  proper  order  and  thus  to  mentally  recon- 
struct the  embryo. 

CONSERVATION  OF  MATTER  AND  ENERGY 

Again  making  use  of  a  cube  passing  obliquely 
downward  thru  a  plane  (as  in  next  figure)  we  note 


Sections  of  Bodies  in  One  Space         79 


that  the  section  of  the  cube  made  by  the  plane  is 
continually  changing  in  magnitude.  Starting  as 
a  point  (when  the  lowest  corner  of  the  cube  en- 
ters the  plane)  the  section  changes  into  an  increas- 
ing triangle,  then  into  polygons  of  more  than 
three  sides,  then  into  a  decreasing  triangle  which 
finally  contracts  into  a  point  (as  the  highest  cor- 
ner of  the  cube  emerges  from  the  plane). 

The  Flatlanders  living  in  that  plane  would  ob- 
serve that  the  area  of  this  section  is  continually 
changing,  i.e.,  this  area  does  not  remain  constant. 
This  suggests  the  possibility  that  the  total  mass 
of  our  material  universe  (considered  as  the  sec- 
tion of  a  higher-dimensional  configuration  passing 
thru  our  space  of  three  dimensions)  may  in  like 
manner  be  continually  changing,  i.e.,  the  total 
mass  of  our  material  universe  may  not  be  constant 
and  hence  the  principle  of  the  conservation  of 
matter  may  not  hold  true.  And  this  may  apply 
not  only  to  our  material  universe  as  a  whole  but 
also  to  ail  of  its  parts  from  the  smallest  atom 
to   the   largest  planets   and   suns.    To   illustrate 


8o     The  Fourth  Dimension  and  the  Bible 

again,  suppose  that  instead  of  the  cube  an  auto- 
mobile is  passing  downward  thru  the  plane.  The 
plane  section  of  the  automobile  will  be  of  very 
great  complexity,  and  not  only  the  whole  of  the 
section  but  also  all  of  its  many  parts  would  be 
continually  changing  in  form  and  area.  And  not 
only  that,  but  during  the  interval  between  the 
instant  when  a  section  of  the  automobile  first  ap- 
peared in  the  plane  and  the  instant  the  last  trace 
of  it  vanished  the  particular  sections  of  hundreds 
of  the  individual  parts  of  the  automobile  would 
suddenly  appear  (be  created),  pass  thru  a  con- 
tinuous series  of  changes,  and  then  as  suddenly 
vanish  (be  annihilated).  For  instance,  we  can 
visualize  the  section  of  some  particular  bolt  in  the 
automobile  as  it  is  passing  thru  the  plane. 

May  it  not  be  possible  that  the  whole  of  our 
material  universe  as  well  as  all  its  component  parts 
are  similarly  changing  in  form  and  total  mass  so 
that  there  may  be  additions  to  or  subtractions 
from  this  total  which  we  as  three-dimensional 
beings  cannot  account  for?  Astronomers  have 
observed  the  sudden  appearance  in  the  heavens 
of  stars  and  comets,  and  also  the  apparent  van- 
ishing of  such  celestial  bodies,  all  in  a  manner 
suggesting  that  they  were  either  entering  or  leav- 
ing our  universe.  For  these  and  other  reasons 
there  has  been  for  some  time  a  suspicion  in  the 


Sections  of  Bodies  in  One  Space  8i 

minds  of  many  scientists  that  the  total  mass  of 
our  material  universe  is  not  constant.  Since  every 
atom  of  matter  is  a  reservoir  of  energy  it  follows 
that  if  the  principle  of  the  conservation  of  matter 
is  not  true,  then  the  principle  of  the  conservation 
of  energy  also  fails.  Numerous  instances  are  re- 
corded in  the  Bible  where  new  matter  or  new 
energy  was  apparently  added  to  our  material 
universe  by  supernatural  means.  The  manna  fall- 
ing in  the  wilderness,  the  ever  replenished  oil  in 
the  widow's  cruse,  and  the  loaves  and  fishes  which 
fed  the  five  thousand,  all  this  food  may  have 
been  the  three-dimensional  sections  of  higher- 
dimensional  supplies  projected  at  those  particular 
times  thru  our  space  by  some  higher  agency.  Also 
the  energy  which  divided  the  waters  of  the  Red 
Sea,  which  caused  the  walls  of  Jericho  to  fall, 
and  by  means  of  which  Jesus  stilled  the  storm  at 
sea  and  walked  on  its  waters,  this  energy  was 
in  some  to  us  mysterious  way  added  to  the  already 
existing  energy  in  our  universe.  When  God 
translated  Enoch  and  Elijah  bodily  into  heaven 
matter  and  energy  was  subtracted  from  our 
material  universe. 


CHAPTER   VIII 

THE  ILLUSORY  AND  THE  REAL 

In  the  case  of  the  cube  passing  thru  a  plane  it 
is  evident  that  the  cube  itself  is  the  real,  the  true 
individual,  while  the  sections  (squares,  triangles, 
etc.)  of  it  seen  by  the  Flatlanders  living  in  that 
plane  are  but  illusory,  inadequate  images  of  the 
cube.  However,  it  is  only  thru  these  countless 
sections  that  the  cube  is  manifested  to  the  Flat- 
landers.  But  the  "birth"  of  a  section  when  the 
cube  enters  the  plane  is  not  the  "birth"  of  the 
cube,  and  the  "death"  of  a  section  when  the  cube 
leaves  the  plane  is  not  the  "death"  of  the  cube; 
this  "birth"  and  this  "death"  are  mere  illusions 
as  far  as  the  cube  itself  is  concerned.  And  so, 
because  man  is  a  higher-dimensional  being  his 
birth,  that  is,  his  entrance  into  this  world,  is 
not  his  beginning,  nor  will  his  death,  that  is,  his 
passing  out  of  this  world,  be  his  end. 

The  classical  allegory  of  the  chained  shadow- 
watchers  given  by  Plato  in  the  seventh  book  of 
his  "Republic"  vividly  portrays  the  relation  be- 
tween true  being  and  the  illusions  of  the  sense 

82 


The  Illusory  and  the  Real  83 

world.  Plato  imagines  a  group  of  prisoners 
chained  just  inside  the  entrance  to  a  large  cavern. 
All  movement  is  impossible  to  them,  they  cannot 
even  turn  their  heads,  hence  they  always  look 
upon  the  opposite  wall  of  the  cavern.  A  strong 
light  streams  in  thru  the  mouth  of  the  cavern. 
The  prisoners  never  see  anything  but  their  own 
shadows  on  the  wall  together  with  the  shadows 
of  all  persons  and  objects  moving  behind  them. 
In  the  course  of  time  they  would  identify  them- 
selves with  their  shadows,  they  would  refer  all 
their  experiences  to  these  shadows,  they  would 
practically  become  the  denizens  of  a  shadow 
world.  All  movements  observed  by  them  would 
be  the  movements  of  shadows  on  the  wall,  the 
movements  of  shapes  with  outlines  but  with  no 
substantiality.  Plato  thus  conceived  of  a  pos- 
sible state  in  which  man  is  reduced  by  his  limited 
consciousness  to  less  than  he  really  is  and  thus 
cleared  the  way  for  the  concept  that  the  normal 
man  in  an  analogous  manner  is  reduced  by  his 
consciousness  to  less  than  he  really  is.  Man  is 
thus  but  the  shadow  cast  on  our  world  by  his  real, 
his  higher  self  dwelling  in  supernal  regions.  As 
the  real  world  of  the  chained  prisoners  was  to 
their  shadow  world,  so  is  the  supernal  world  to 
the  world  of  our  experience. 

Let  us  suppose  the  surface  of  a  smooth  wet 


84     The  Fourth  Dimension  and  the  Bible 


sidewalk  to  be  part  of  Flatland.  A  man  walking 
on  it  would  make  tracks  as  shown  above.  The 
Flatlanders  living  in  the  plane  of  the  surface  of 
the  sidewalk  would  see  only  the  bottoms  of  the 
heels  and  soles  of  his  shoes.  First  the  heel  and 
sole  of  one  foot  would  appear  to  them  apparently 
out  of  nowhere  and  then  vanish  just  as  the  heel 
and  sole  of  the  other  foot  appeared  and  in  turn 
vanished.  This  would  be  repeated  again  and 
again  to  the  great  mystification  of  the  Flatlanders. 
To  them  it  would  seem  as  if  two  two-dimensional 
bodies  (the  bottoms  of  the  heel  and  sole  of  one 
shoe)  were  traveling  together  for  some  unknown 
reason  accompanied  by  their  twins  (the  bottoms 
of  the  heel  and  sole  of  the  other  shoe).  The 
Flatlanders  could  not  conceive  of  each  heel  and 
sole  being  connected,  that  is,  as  belonging  to  the 
same  shoe  (a  three-dimensional  body),  and  much 
less  could  they  conceive  of  these  shoes  as  belong- 
ing to  the  same  three-dimensional  being  (the 
man).  Suppose,  however,  that  some  Flatlander 
of  superior  intellect  had  an  inkling  of  the  third 
dimension  such  as  we  have  of  a  fourth  dimension, 
and  that  he  suspected  that  the  two  pairs  of  impres- 


The  Illusory  and  the  Real  85 

sions  made  on  the  sidewalk  really  belonged  to  the 
same  three-dimensional  being.  As  only  the  bot- 
toms of  the  man's  heels  and  soles  would  come 
within  his  range  of  observation  he  would  prob- 
ably state  as  his  opinion  that  this  three-dimen- 
sional being  (the  man)  had  no  intelligence  or 
feeling  and  therefore  was  of  little  or  no  account. 
He  would  judge  the  man  from  what  he  had  seen 
of  the  bottoms  of  his  heels  and  soles !  By  analogy 
then,  if  man  is  a  higher-dimensional  being  of 
whom  we  can  observe  his  three-dimensional 
attributes  only,  it  may  be  that,  comparatively 
speaking,  we  have  no  clearer  or  fuller  conception 
of  what  man  actually  Is  in  the  sight  of  God  than 
had  the  Flatlander  when  he  tried  to  describe  a 
three-dimensional  being  (the  man)  from  what  he 
had  seen  of  the  bottoms  of  his  heels  and  soles. 


CHAPTER   IX 

SPACES  AS  HEAVENS  AND  HELLS 

Perhaps  the  higher-dimensional  spaces  are  the 
heavens  to  which  we  ascend  at  death  if  we  have 
so  lived  on  earth  as  to  deserve  such  promotion; 
and  the  lower-dimensional  spaces  are  the  hells  to 
which  we  will  be  consigned  in  case  our  lives  have 
fallen  below  a  certain  standard.  Thus  at  death 
the  soul  of  a  good  man  (his  higher-dimensional 
self)  shakes  off  the  limitations  of  understanding, 
knowledge  and  movement  which  our  three-dimen- 
sional space  imposed  on  him,  and  enters  space  of 
four  dimensions.  Instead  of  being  restricted  to 
three  degrees  of  freedom  of  motion  he  will  then 
be  able  to  move  in  four  independent  directions. 
Such  would  be  his  increased  freedom  of  action 
that,  if  he  could  look  back,  his  former  life  in 
our  space  would  probably  appear  to  him  as  life 
in  a  prison  cell  would  now  seem  to  us.  Unlimited 
vistas  of  new  knowledge  would  open  up  to  him, 
and  his  powers  of  understanding  and  possibilities 
of  action  would  have  increased  to  an  extent  far 
beyond  the  present  limits  of  human  conception. 

86 


spaces  as  Heavens  and  Hells  87 

Coming  from  the  restrictions  imposed  by  our 
space,  the  space  of  four  dimensions  would  indeed 
seem  heaven  to  him !  On  the  other  hand,  the  soul 
of  the  wicked  man  would  at  death  be  condemned 
to  live  in  Flatland,  the  space  of  two  dimensions. 
He  could  move  only  in  two  independent  directions, 
all  his  knowledge  and  experience  of  a  third  dimen- 
sion would  be  blotted  out  of  his  mind,  and  life  in 
Flatland  would  certainly  be  hell  to  him.  If  during 
his  existence  in  Flatland  the  fallen  man  did  not 
mend  his  ways  but  continued  to  deteriorate  in 
character,  then  at  death  he  would  again  be  de- 
moted, he  would  be  consigned  to  Lineland,  an 
experience  that  would  abound  with  horrors.  He 
could  then  move  in  one  direction  only  and  his  fund 
of  knowledge  and  his  powers  of  understanding 
would  be  still  further  diminished.  If  after  this 
awful  experience  he  still  persisted  in  his  evil  ways 
he  would  at  death  be  consigned  to  space  of  zero 
dimensions,  that  is,  to  a  point.  There  no  motion 
at  all  would  be  possible,  all  his  knowledge  and 
understanding  would  have  vanished,  he  would  be 
in  the  deepest  pit  of  hell.  "For  a  fire  is  kindled 
in  mine  anger,  and  shall  burn  unto  the  lowest 
hell."     Deut.  32  :22. 

But  let  us  turn  back  to  the  more  pleasant  pros- 
pect, the  assumption  that  our  earthly  friend  has 
so  lived  that  at  death  he  is  translated  into  space 


88      The  Fourth  Dimension  and  the  Bible 

of  four  dimensions,  that  is,  the  first  heaven  for 
us.  "And  I  saw  a  new  heaven  and  a  new 
earth:  for  the  first  heaven  and  the  first  earth 
were  passed  away."  Rev.  21  :i.  Happily  during 
his  life  in  space  of  four  dimensions  our  friend 
continues  his  striving  after  that  which  is  highest 
and  noblest  and  best,  and  so  when  he  has  finished 
his  course  there  he  Is  promoted  to  space  of  five 
dimensions,  that  is,  our  second  heaven,  a  place 
of  far  surpassing  knowledge  and  freedom  and 
glory.  Continuing  in  growth  of  character  he 
would  then  be  advanced  to  space  of  six  dimen- 
sions, that  Is,  our  third  heaven.  "I  knew  a  man 
in  Christ  above  fourteen  years  ago  (whether  in 
the  body  I  cannot  tell;  or  whether  out  of  the 
body,  I  cannot  tell:  God  knoweth),  such  an  one 
caught  up  into  the  third  heaven."  II  Cor.  12:2. 
Not  stopping  here  we  can  Imagine  that  the 
noblest  and  greatest  individuals  will,  after  such 
successive  existences,  be  translated  to  still  higher 
spaces  (heavens).  The  higher  the  order  of  space 
the  greater  and  more  glorious  would  be  the  state 
of  the  blessed  who  dwell  there,  until  the  space  of 
Infinite  dimensions  is  reached  where  God  has  his 
dwelling  place.  There  are  no  restrictions  to 
motion  and  no  limitations  of  power,  knowledge 
or  understanding.  God  Is  omnipotent,  omniscient, 
omnipresent;  the  very  personification  of  justice 


spaces  as  Heavens  and  Hells  89 

and  mercy,  wisdom  and  love.  "God  is  absolutely 
infinite,  consisting  of  infinite  attributes  each  ex- 
pressing eternal  and  infinite  essentiality." 

The  Bible  has  made  the  above  concept  of  more 
than  one  heaven  familiar  to  us  all.  "In  my 
Father's  house  are  many  mansions."  John  14:2. 
"The  heaven,  even  the  heavens,  are  the  Lord's." 
Ps.  115  :i6.  "When  I  consider  thy  heavens."  Ps. 
8  :3.  "When  he  prepared  the  heavens."  Prov. 
8  :27.  "Therefore  will  I  shake  the  heavens."  Isa. 
13.3.  "For,  behold,  I  create  new  heavens."  Isa. 
65:17.  Also  Hag.  2:6,  Ezek.  1:1,  Matt.  3:16, 
Mark  1:10,  II  Cor.  5:1,  II  Peter  3:13.  We 
have  referred  to  higher  and  lower  orders  of 
spaces.  That  there  are  also  various  orders  of 
heavens  may  be  inferred  from  scripture  passages. 
"Behold,  the  heaven  and  the  heaven  of  heavens 
is  the  Lord's  thy  God."  Deut.  10:14.  "Behold, 
the  heaven  and  the  heaven  of  heavens  cannot  con- 
tain thee."     I  King  8  :27. 

We  find  nothing  in  the  hypothesis  of  higher 
spaces  that  we  have  been  developing  which  con- 
tradicts the  reality  of  sin  or  the  fact  of  the  divine 
redemption  of  a  fallen  race.  Whether  or  not 
the  inhabitants  of  other  spaces  than  ours  have 
passed  or  are  passing  thru  the  experiences  of  an 
Adam's  fall  and  a  Christ's  atonement  we  do  not 
know,  but  the  presumption  is  strong  that  at  least 


90     The  Fourth  Dimension  and  the  Bible 

some  of  the  dwellers  in  higher  space  (heavens) 
did  disobey  God  and  therefore  incurred  the  in- 
evitable punishment  which  is  sure  to  overtake  the 
sinner,  whether  he  be  man  or  angel.  "Behold,  he 
putteth  no  trust  in  his  saints :  yea,  the  heavens 
are  not  clean  in  his  sight."  Job  15  :i5.  "For  if 
God  spared  not  the  angels  that  sinned  but  cast 
them  down  to  hell,  and  delivered  them  into  chains 
of  darkness,  to  be  reserved  unto  judgment."  II 
Peter  2  14.  "And  the  angels  which  kept  not  their 
first  estate,  but  left  their  own  habitation,  he  hath 
reserved  in  everlasting  chains  under  darkness  unto 
the  judgment  of  the  great  day."  Jude  6.  "And 
he  said  unto  them,  I  beheld  Satan  as  lightning  fall 
from  heaven."    Luke  10:18. 


CHAPTER  X 

CREATION    OF   OUR    MATERIAL   UNL 
VERSE.  EVOLUTION 

Is  the  theory  of  higher  spaces  in  accord  with 
the  Bible  account  of  creation?  Let  us  approach 
this  subject  by  making  use  of  the  concept  em- 
ployed in  the  last  chapter,  the  assumption  that 
bodies  in  any  given  space  are  the  sections  of 
higher-dimensional  bodies  made  by  that  space. 
For  example,  let  the  plane  AB  represent  a  portion 
of  a  two-dimensional  space  (Flatland). 


Assume  figure  (a)  to  be  a  solid  circular  cylinder 
(three-dimensional  body)  with  its  axis  always 
perpendicular  to  the  plane  and  moving  towards 

91 


92     The  Fourth  Dimension  and  the  Bible 

and  completely  thru  the  plane  and  assume  the 
rest  of  the  figures  as  successive  stages  of  the  same 
cylinder  as  it  is  passing  thru  the  plane.  Before 
the  cylinder  reaches  the  plane,  Flatlanders  living 
in  that  plane  would  know  absolutely  nothing 
about  it,  the  cylinder  would  be  non-existent  as  far 
as  they  are  concerned.  The  instant  the  lower  base 
of  the  cylinder  coincided  with  the  plane,  as  in  fig- 
ure (Z?),  the  circle  CD,  the  section  of  the  cylinder 
made  by  the  plane,  would  suddenly  appear  before 
the  astonished  Flatlanders,  coming  apparently 
out  of  nowhere.  To  them  this  phenomenon  would 
be  the  creation  of  a  circle  {two-dimensional  body) 
out  of  nothing.  They  could  no  more  account  for 
this  miracle  than  we  can  account  for  the  creation 
of  our  material  {three-dimensional)  universe  out 
of  nothing.  They  could  not  even  imagine  an  ex- 
planation of  the  mystery,  they  could  simply  state 
that  where  there  was  nothing  before  there  was 
now  a  circle;  a  real  two-dimensional  body  had 
been  created  apparently  out  of  nothing. 

During  the  time  the  cylinder  is  passing  thru 
their  plane  the  Flatlanders  would  behold  the  cir- 
cular sections  made  by  the  plane  as  a  continuing 
entity,  a  circle,  (as  EF,  GH,  etc.)  until  its  upper 
base  KL,  figure  (/),  coincided  with  the  plane. 
Then  the  instant  the  cylinder  passed  out  of  the 
plane  the  circle  would  suddenly  vanish,  apparently 


Creation  of  Our  Material  Universe      93 

into  nowhere.  To  the  Flatlanders  the  circle  would 
cease  to  exist,  where  there  had  been  something 
before  there  was  now  nothing.  They  could  no 
more  account  for  the  annihilation  of  the  circle 
than  we  can  conceive  of  the  possibility  of  the  an- 
nihilation of  our  material  universe,  it  would  be 
and  remain  an  unsolvable  mystery  to  them. 

Continuing  this  line  of  reasoning  one  step  fur- 
ther we  may  think  of  our  material  universe  as 
the  section  of  a  higher-dimensional  body  (aggre- 
gate) made  by  our  space  of  three  dimensions. 
Just  as  the  circle,  in  what  has  gone  before,  was 
a  two-dimensional  body  created  by  the  entrance 
of  a  cylinder  (three-dimensional  body)  into  two- 
dimensional  space  (Flatland)  so  we  may  conceive 
of  our  material  universe  as  being  created  by  the 
entrance  of  a  higher-dimensional  body  (aggre- 
gate) into  our  three-dimensional  space.  As  human 
beings  restricted  to  the  knowledge  and  experi- 
ences of  a  three-dimensional  space  we  can  have 
no  knowledge  of  how  this  can  be  brought  about 
or  understand  the  real  nature  of  the  higher-dimen- 
sional body  (aggregate)  in  question.  All  that 
we  can  say  is  that  where  there  was  nothing  before 
we  now  behold  our  material  universe,  that  is,  our 
material  universe  was  created  out  of  nothing. 
"In  the  beginning  God  created  the  heaven  and 
the  earth."    Gen.  1:1.   "Through  faith  we  under- 


94     The  Fourth  Dimension  and  the  Bible 

stand  that  the  worlds  were  framed  by  the  word 
of  God,  so  that  things  which  are  seen  were  not 
made  of  things  which  do  appear.'*  Heb.  11:3. 
Just  as  Flatlanders  (p.  95)  continued  to  behold 
as  a  continuing  entity  the  circular  section  of  the 
cylinder  made  by  their  space  (plane)  while  the 
cylinder  was  passing  thru  the  plane,  so  we  today 
behold  our  material  universe  as  a  continuing  en- 
tity, an  aggregate  of  planets,  stars,  and  nebulas, 
which  aggregate  we  may  conceive  of  as  being  the 
section  of  a  higher-dimensional  configuration  of 
surpassing  greatness  and  variety  formed  by  the 
passing  of  this  transcendent  configuration  thru  our 
space  of  three  dimensions.  If,  in  our  illustration, 
instead  of  a  circular  cylinder  some  irregular 
shaped  solid  had  been  passed  thru  the  plane  then 
the  section  seen  by  the  Flatlander  would  have 
been,  not  a  circle,  but  a  continually  changing  irre- 
gular shaped  plane  figure  ( two-dimensional  body) . 
And  so,  because  every  part  of  our  material  uni- 
verse from  the  smallest  particle  of  matter  on 
earth  to  the  most  distant  star  is  in  a  state  of 
continual  change  we  may  assume  that  these 
changes  are  merely  three-dimensional  manifes- 
tations of  a  transcendental  configuration  of  in- 
finite richness  in  its  composition  and  form  caused 
by  the  passing  of  the  latter  thru  our  space  of 
tliree   dimensions.     According  to   this  view  the 


Creation  of  Our  Material  Universe      95 

events  of  the  six  days  of  creation  as  recorded  in 
the  Bible,  as  well  as  all  other  events  which  have 
since  taken  place  and  are  at  this  moment  taking 
place,  are  three-dimensional  sections  (images, 
manifestations)  of  some  higher-dimensional  ag- 
gregate (heaven)  in  which  God  dwells. 

In  the  case  of  the  cylinder  and  the  plane  we 
pointed  out  on  pp.  92  and  97  that  as  the  cylinder 
passed  out  of  the  plane  the  circle  which  the  Flat- 
landers  had  observed  suddenly  vanished  before 
their  eyes  after  its  coincidence  with  the  upper  base 
of  the  cylinder,  and  was  seen  by  them  no  more. 
No  trace  whatever  of  the  circle  remained  in  the 
two-dimensional  world  of  the  Flatlanders,  to  them 
it  appeared  as  the  total  annihilation  of  a  body 
(the  circle)  in  their  space (  plane),  a  phenomenon 
which  they  could  in  no  way  account  for.  The 
Flatlanders  might  well  record  it  in  their  annals 
as  the  end  of  the  circle.  And  so  to  us  (three- 
dimensional  beings)  the  end,  the  total  annihi- 
lation, of  our  material  three-dimensional  uni- 
verse may  be  considered  as  coming  to  pass  when 
the  higher-dimensional  (transcendent)  configura- 
tion to  which  we  have  referred  passes  out  of  our 
three-dimensional  space.  Numerous  Bible  pas- 
sages indicate  unmistakably  that  not  only  did  our 
universe  have  a  beginning  (was  created)  but  that 
it  will  also  in  the  fullness  of  time  have  an  end 


g6     The  Fourth  Dimension  and  the  Bible 

(be  annihilated)  Ps.  75:3,  Mat.  13:40,  II  Pet. 
3:10,  13,  Rev.  21:1.  The  Bible  does  not,  and 
could  not,  tell  us  how  something  can  become  noth- 
ing any  more  than  it  can  tell  us  how  nothing  can 
become  something  because  it  was  written  for  hu- 
man (three-dimensional)  beings  wholly  incapable 
of  comprehending  such  (higher-dimensional) 
mysteries. 

EVOLUTION 

Let  us  assume  a  solid  circular  cone   (three-di- 
mensional body)    passing  thru  a  plane,  as  AB, 


a 


V- 


V 


(two-dimensional  space)  the  axis  of  the  cone  being 
always  perpendicular  to  the  plane. 

Before  the  cone  reaches  any  part  of  the  plane, 
as  in  the  position  fig.  («),  the  cone  is  non-exist- 
ent to  the  Flatlanders  living  in  the  plane.  The 
instant  the  apex  of  the  cone  enters  the  plane  (fig. 
(Z?),  the  point  C  (the  section  of  the  cone  made 
by  the  plane)   will  appear  to  the  Flatlanders  in 


Evolution  97 

that  plane  coming  apparently  from  nowhere.  To 
them  it  will  be  the  creation  of  a  point  out  of 
nothing.  Then  as  the  cone  continues  to  move 
thru  the  plane  this  section  (point)  will  be  seen  to 
develop  into  a  circle  (as  DE)  with  a  continually 
increasing  radius.  In  other  words,  the  Flat- 
landers  would  behold  the  evolution  of  a  point  into 
a  circle.  And  this  evolutionary  process  would 
continue  until  the  circular  section  has  grown  into 
the  circle  FG,  the  base  of  the  cone.  As  the  cone 
passes  out  of  the  plane  the  circle  will  vanish,  ap- 
parently into  nowhere.  A  Flatlander  living  in 
the  plane  AB  and  observing  all  that  had  taken 
place  might  well  describe  it  all  as  the  creation,  the 
development  by  evolution,  and  the  annihilation 
of  a  circle;  and  this  without  having  any  concep- 
tion whatever  of  the  existence  of  the  cone  itself 
or  its  relation  to  the  phenomenon  he  had  wit- 
nessed. 

May  it  not  be  that  all  evolutionary  processes 
in  our  material  three-dimensional  universe  are 
in  like  manner  three-dimensional  sections  (images, 
manifestations)  of  the  elements  of  some  higher 
dimensional  (transcendent)  aggregate  of  infinite 
complexity?  In  this  concept  there  is  no  conflict 
between  the  doctrine  of  instantaneous  creation 
and  the  theory  of  evolution,  each  is  a  part  of  the 
process  involved.     Both  of  them,  however,  ope- 


98     The  Fourth  Dimension  and  the  Bible 

rate  In  a  manner  incomprehensible  to  three-dimen- 
sional beings,  both  are  miracles  to  us.  To  God 
instantaneous  creation  and  evolutionary  processes 
are  equally  possible,  he  can  and  does  operate  with 
either  or  both  according  to  his  Infinite  wisdom. 
To  deny  the  possibility  of  Instantaneous  creation 
Is  to  deny  the  power  of  God  and  to  cast  over- 
board the  concept  of  evolution  is  to  deny  the 
existence  of  the  numberless  examples  of  evolution- 
ary processes  which  are  developing  today  before 
our  very  eyes  or  are  unfolded  to  our  minds. 


CHAPTER  XI 

RAISING  OF  THE  DEAD.    SACRAMENT 
OF  THE  LORD'S  SUPPER 

The  soul  of  man,  being  higher-dimensional, 
can  leave  the  human  body,  which  is  three-dimen- 
sional, without  doing  any  violence  to  the  body  or 
leaving  any  trace  on  It  of  a  point  of  emergence. 
Investigators  have  often  tried  to  photograph  the 
departing  soul  of  a  dying  man,  and  to  discover 
any  lesions  on  his  body  caused  thereby,  but  always 
without  success.  No  image  of  the  soul,  which  is 
higher-dimensional,  can  be  secured  by  material 
(three-dimensional)  means;  nor  can  the  human 
body  imprison  the  soul  any  more  than  (as  we 
have  pointed  out)  it  would  be  possible  to  imprison 
a  four-dimensional  being  in  a  three-dimensional 
prison  cell  or  to  confine  a  three-dimensional  being 
within  two-dimensional  (Flatland)  barriers. 

Likewise  it  follows  that  a  soul  may  enter  a 
human  body  without  hindrance  as  far  as  spacial 
considerations  are  concerned,  just  as  a  three-di- 
mensional being  may  freely  enter  a  closed  com- 
partment in  Flatland.     The  raising  of  the  dead 

99 


100  The  Fourth  Dimension  and  the  Bible 

involves  the  re-entrance  of  souls  into  their  for- 
mer corporeal  habitations.  What  the  forces  are 
which  can  bring  this  about  is  a  deep  mystery  to 
us.  We  know,  however,  that  Jesus  had  the  power 
to  perform  this  miracle,  and  that  thru  him,  also 
Paul  and  Peter  raised  the  dead.  Jesus  restored 
to  life  the  daughter  of  Jairus  (Mat.  9:16:  Mark 
5  14:  Luke  8  :54),  the  son  of  the  widow  of  Nain 
(Luke  7:15)  and  Lazarus  (John  11),  who  had 
been  four  days  in  his  grave.  Paul  did  the  same 
for  Eutychus  (Acts  20:10)  and  Peter  the  same 
for  Tabitha  (Acts  9:40).  Likewise  Elijah  (I 
King  17:17,  23)  and  Elisha  (II  King  4:32,  37) 
raised  the  dead.  The  dead  came  out  of  their 
graves  on  the  first  Good  Friday,  later  entering 
the  Holy  City  and  appearing  unto  many.  (Mat. 
27:52). 

That  we  have  no  record  of  the  experiences  of 
Lazarus  and  the  others  who  were  raised  from  the 
dead,  covering  the  interval  between  their  deaths 
and  their  coming  to  life  again,  has  been  a  keen 
disappointment  to  many  Bible  readers.  In  fact, 
this  seeming  omission  of  a  description  of  the  de- 
tails of  their  existence  during  this  death  interval 
has  been  used  by  Bible  scoffers  not  only  to  cast 
doubt  on  the  truth  of  these  miracles  but  also  to 
discredit  all  Christian  beliefs.  Where  was  the 
immortal  part  of  Lazarus  during  the  four  days 


Raising  of  the  Dead  loi 

In  which  his  body  was  in  the  grave,  what  did  he 
do,  what  were  his  experiences?  A  message  from 
Mars  would  be  of  trivial  interest  to  dwellers  on 
earth  compared  with  the  importance  of  the  mes- 
sage that  Lazarus  should  have  brought  back  from 
the  regions  beyond  the  grave.  It  would  have  an- 
swered the  great  burning  question  of  past,  pres- 
ent and  future  ages,  the  question  beside  which 
all  other  questions  vanish  into  insignificance.  That 
Lazarus  did  not  answer  this  question  is  evident 
because  if  he  had  it  would  have  been  recorded. 
The  people  of  his  day  were  no  less  consumed  by 
interest  in  the  answer  to  this  question  than  we 
are  today.  If  Lazarus  had  related  his  experi- 
ences during  those  four  fateful  days  his  narrative 
certainly  would  have  been  handed  down  to  us  in 
some  form. 

The  reason  why  Lazarus  did  not  answer  this 
question,  did  not  relate  his  experiences  between 
his  death  and  his  return  from  the  grave,  was  be- 
cause he  could  not.  Granting  that  he  had  a  con- 
scious existence  during  this  four-day  period  it 
may  have  been  that  of  his  higher-dimensional 
being  (his  soul,  his  spiritual  self)  in  a  space  of 
higher  dimensions,  a  space  radically  different 
from  our  space  of  three  dimensions.  The  im- 
pressions he  received  were  therefore  the  impres- 
sions received  in  a  space  totally  foreign  to  us, 


102   The  Fourth  Dimension  and  the  Bible 

Impressions  which  could  not  be  described  to  us 
because  we  have  no  words,  no  language  which 
will  convey  such  a  description  to  human  (three- 
dimensional)  beings.  What  Is  more,  any  such 
impressions  that  Lazarus  may  have  been  con- 
scious of  before  Jesus  raised  him  from  the  grave 
vanished  the  instant  he  came  to  life,  that  is,  the 
Instant  he  again  became  a  mortal,  a  three-dimen- 
sional being.  On  being  again  clothed  with  his 
material  body  his  spirit  at  once  became  subject  to 
the  restrictions  and  limitations  imposed  on  be- 
ings living  in  our  space  of  three  dimensions. 

It  seems  futile  therefore  to  expect  that  we  shall 
ever  receive  a  message  thru  human  agencies  from 
those  who  have  departed  to  the  great  beyond  de- 
scribing their  state  of  existence  because  such  a 
message  can  convey  no  meaning  whatever  to  us. 
The  Bible  contains  no  promise  of  any  such  knowl- 
edge but  on  the  contrary  asserts  that  this  must  re- 
main a  sealed  book  to  mortals.  This  does  not, 
however.  Interpose  any  obstacles  whatever  to  di- 
vine revelations  to  man,  either  In  the  past  or  In 
the  future,  because  God  is  not  limited  in  any  man- 
ner by  the  restrictions  of  any  space. 

THE    SACRAMENT   OF   THE    LORD^S   SUPPER 

All  the  different  shades  of  doctrine  held  by 
Christian  believers  regarding  the  sacrament  of 


Sacrament  of  the  Lord's  Supper        103 

the  Lord's  Supper  may  be  roughly  divided  into 
three  distinct  groups. 

Group  I.  Doctrines  which  include  transubstan- 
tiation  as  a  fundamental  principle.  These  assert 
that  the  bread  and  wine  are  literally  transformed 
into  the  body  and  blood  of  Jesus.  According  to 
this  principle  of  transubstantiation  the  believer 
partakes,  not  of  the  bread,  but  of  the  actual  ma- 
terial human  body  of  Jesus,  and  in  drinking  the 
cup  drinks,  not  wine,  but  the  actual  material  human 
blood  of  Jesus. 

Group  II.  Doctrines  which  hold  that  the  bread 
and  wine  only  represent  or  symbolize  the  body 
and  blood  of  Jesus.  According  to  these  the  be- 
liever partakes  only  of  the  material  bread  and 
wine,  the  act  being  merely  a  remembrance,  a  cele- 
bration commemorating  the  last  supper  of  Jesus 
with  his  disciples. 

Group  III.  Doctrines  which  assert  that  the  be- 
liever in  partaking  of  the  bread  and  wine  also 
by  the  same  act  partakes  not  of  the  material 
body  and  blood  of  Jesus  but  of  the  spiritual  body 
and  blood  of  a  resurrected,  a  glorified  Jesus. 

The  first  of  these  three  groups  of  doctrines  is 
naturally  repellant  to  us  because  of  the  grossness 
of  the  act  involved,  namely,  the  consumption  of 
human  flesh  and  blood.  It  is  the  human  body  and 
blood  of  Jesus,  into  which  the  bread  and  wine 


I04   The  Fourth  Dimension  and  the  Bible 

has  been  transformed,  of  which  the  believer  is 
supposed  to  partake. 

On  the  other  hand,  the  second  of  these  three 
groups  of  doctrines  does  not  entirely  satisfy  those 
who  are  spiritually  minded  because  it  minimizes 
the  sacrament  by  making  it  merely  a  commemora- 
tion ceremony.  This  viewpoint  is  difficult  to  rec- 
oncile with  the  words  of  Jesus,  "this  is  my  body 
.  .  .  this  is  my  blood." 

Defenders  of  the  doctrine  of  transubstantiation 
invariably  attempt  to  justify  their  belief  in  it  by 
quoting  the  above  words  of  Jesus.  In  so  doing 
they  are  trying  to  support  their  case  on  the  wholly 
unwarranted  assumption  that  Jesus  spoke  of  his 
material,  his  human  body  and  blood.  Jesus  knew 
of  his  coming  suffering,  death  and  resurrection. 
He  spoke  of  the  future  beyond  Golgotha  when 
he  said  "this  do,  as  oft  as  ye  drink  it,  in  remem- 
brance of  me."  In  this  future  there  would  be 
no  more  a  material,  a  human  body  and  blood  of 
Jesus,  but  only  his  spiritual,  his  glorified  body 
and  blood. 

The  third  of  the  above  groups  of  doctrines 
therefore  is  sanctioned  by  the  words  of  Jesus. 
It  also  commends  itself  to  human  reason.  It  is 
therefore  near  at  hand  to  assert  that  Jesus  is 
present  in  the  sacrament  of  the  Lord's  Supper 
as    a    higher-dimensional    being.     He    descends 


Sacrament  of  the  Lord's  Supper        105 

from  heaven  (higher-dimensional  space)  to  com- 
mune with  the  believer  in  our  world  (three-dimen- 
sional space).  We  cannot  detect  his  presence 
thru  our  material  senses  because  he  is  a  higher- 
dimensional  being  while  we  are  laboring  under  the 
restrictions  and  limitations  imposed  on  three-di- 
mensional beings.  There  is  no  necessity  for  the 
spacial  presence  (in  the  ordinary  sense)  of  Jesus 
in  order  to  fulfill  his  promise  to  be  with  us.  How 
much  greater  and  glorious  as  well  as  more  satis- 
fying to  the  believer  is  not  this  presence  of  the 
spiritual,  the  glorified  (higher-dimensional)  body 
and  blood  of  Jesus  than  would  be  the  presence  of 
merely  his  human,  his  material  body  and  blood! 


CHAPTER  XII 

DESCRIPTION  OF  HEAVEN.    MIRACLES 

It  is  evident  that  we  as  three-dimensional  be- 
ings cannot  have  a  true  conception  of  what  heaven 
is;  being  a  higher  space  it  is  beyond  our  powers 
of  understanding.  If  an  accurate  description  of 
heaven  were  in  some  way  available  we  would 
not  understand  an  iota  of  it,  it  would  be  mean- 
ingless to  us.  "For  since  the  beginning  of  the 
world  men  have  not  heard,  nor  perceived  by  the 
ear,  neither  hath  the  eye  seen,  O  God,  besides 
thee,  what  he  hath  prepared  for  him  that  wait- 
eth  for  him."  Isa.  64:4.  "Eye  hath  not  seen, 
nor  ear  heard,  neither  have  entered  into  the  heart 
of  man,  the  things  which  God  hath  prepared  for 
them  that  love  Him."  I  Cor.  2  :<).  Christ  spoke 
to  his  Apostles  of  the  impossibility  of  their  hav- 
ing any  image  or  notion  of  the  place  to  which 
when  he  disappeared  he  would  go  and  whence  he 
would  return.  Attempts  to  describe  heaven  al- 
ways result  in  a  description  in  terms  of  our  three- 
dimensional  environment.     In  this  connection  it 

106 


Description  of  Heaven 


107 


is  significant  to  note  that  the  Bible  usually  tells 
what  heaven  is  not  rather  than  what  heaven  is. 
The  adjectives  are  generally  negative.  We  should 
never  lose  sight  of  the  fact  that  the  Bible  was 
written  for  beings  living  in  a  three-dimensional 
space,  beings  whose  knowledge  and  experiences 
are  wholly  limited  to  that  space.  Dr.  P.  Anstadt 
in  his  book  "Recognition  in  Heaven"  gives  the 
following  summary  of  Bible  statements  describ- 
ing heaven : 


Negative  Features,  or 
the  Things  that  Will 
Not  Be  There 

Indestructible. 

Undefilable. 

Unchangeable. 

No  crying. 

No  tears 

No  pain. 

No  sorrow. 

No  death. 

No  burning  sun. 

No  cold  or  heat. 

No  night. 

No  hunger. 

No  thirst. 

No  bad  men. 

No  sin. 

No  curse. 


Positive  Features,  or 
the  Things  that  Will 
Be  There. 

The  city  of  our  God — 

the  heavenly  Jerusalem. 

Beautiful  waters. 

Delicious  fruits. 

Sure  healing  for  the  na- 
tions. 

Populous  with  happy 
people. 

Beautiful   garments. 

Devout  worship. 

Enchanting  music. 

A  just  ruler. 

An  eternal  kingdom. 

The  grandest  capitol. 

Many  mansions. 


io8   The  Fourth  Dimension  and  the  Bible 

MIRACLES 

One  reason  why  some  very  good  people  hesi- 
tate  about   accepting  the   miracles   of   the   Bible 
at  their  face  value  is  that  they  assume  that  such 
miracles  can  only  be  brought  about  by  violating 
or   in   some   way   interfering   with   the   laws   of 
nature,  the  laws  governing  our  three-dimensional 
universe.     This  difficulty  vanishes  if  instead  we 
look  upon  the  miracles  of  the  Bible  as  the  per- 
fectly logical  results  of  the  working  out  of  laws 
connected  with  higher  spaces,  laws  which  include 
the  lower,  the  so-called  natural  laws  of  our  ma- 
terial  universe,   just   as   our   space   is   contained 
within  the  higher  spaces.    We  may  then  consider 
the  miracles  performed  by  Jesus,  the  apostles  and 
the  prophets  to  consist  of  momentary  revelations 
by  God  to  mortals  of  the  operations  of  some  of 
those  higher  laws  of  which  we  are  as  yet  igno- 
rant.    It  is  not  at  all  probable  that  the  apostles 
and  prophets  themselves  understood  these  higher 
laws,  and  it  was  not  necessary  that  they  should, 
any  more  than  that  it  is  necessary  for  the  operator 
who  sends  a  message  by  wireless  to  understand 
the  theory  of  wireless  telegraphy  or  to  have  actual 
knowledge  of  the  medium  thru  which  it  is  trans- 
mitted.    These  higher  laws  may  be  the  laws  of 
four-dimensional   or  still  higher  spaces  or  they 


Miracles  109 

may  be  laws  connected  with  realms  of  thought 
and  being  of  which  we  now  have  no  conception 
whatever.  To  insist  that  God  can  make  wine  out 
of  water  only  by  pouring  fermented  grape  juice 
into  it  is  ridiculous !  If  he  were  thus  limited  he 
would  not  be  God.  The  rending  of  the  veil  of 
the  Temple  on  the  crucifixion  of  Christ  and  the 
speaking  with  many  tongues  on  the  giving  out  of 
the  Holy  Ghost,  these  may  all  be  revelations  to 
man  of  higher-dimensional  phenomena. 

Swedenborg's  involved  descriptions  of  "heav- 
enly" forms,  motions,  and  mechanics,  become 
somewhat  more  intelligible  when  interpreted  in 
terms  of  higher  spaces  even  if  we  are  not  willing, 
as  the  author  is  not,  to  agree  with  his  conclusions. 
Helmholtz  continually  kept  the  possibility  of 
physical  higher  spaces  before  him  in  his  dynami- 
cal studies,  reasonings  which  are  among  the  class- 
ics of  physical  science.  Kelvin  also  felt  the  pres- 
sure of  the  reality  of  higher  spaces  so  strongly 
that  he  declared  himself  ready  to  accept  them 
as  an  explanation  of  physical  phenomena  when 
these  could  be  more  consistently  explained  by  such 
a  concept. 


CHAPTER  XIII 

CONCLUSION 

The  new  vistas  of  thought  which  have  been 
opened  to  us  by  the  higher  space  hypothesis  have 
revealed  unlimited  realms  of  being  and  doing, 
all  of  which  are  interrelated  in  a  manner  far  sur- 
passing the  powers  of  ingenuity  of  even  a  super- 
man. In  the  light  of  this  vision,  how  small  and 
mean,  how  low  and  limited,  yes,  how  contempt- 
ible, does  not  the  materialist  appear  as  he  bur- 
rows into  the  dust  in  his  attempts  to  discover  the 
source  of  the  glorious  rays  of  truth  and  light 
which  radiate  from  God!  As  we,  in  this  study 
of  higher  spaces,  have  caught  a  momentary 
glimpse  of  the  sublimity  and  vastness,  the  glory 
and  grandeur,  the  symmetry  and  order,  of  what 
must  be  only  a  very  small  part,  perhaps  only  an 
infinitesimal  part,  of  creation;  how  preposterous 
does  it  not  seem,  yes,  how  ridiculous  even  to  sup- 
pose that  all  should  have  come  into  being  thru  a 
succession  of  fortuitous  circumstances,  by  blind 
chance,  and  not  thru  a  supreme  intelligence  with 
no  limitations  of  knowledge  or  power. 


Conclusion  1 1 1 

How  pitiful  and  weak  is  man  with  his  limita- 
tions of  thought  and  action  in  the  presence  of 
these  infinite  possibilities  !  And  yet  there  are  men 
presumptuous  enough  to  claim  that  human  reason 
and  understanding  can  account  for  it  all.  No 
matter  how  wise  or  powerful  a  man  may  be  he 
is  but  one  of  billions  of  humans,  to  whom  this 
our  earth  has  been  or  is  the  dwelling  place.  And 
our  earth  is  only  one  of  the  smallest  of  the  plan- 
ets revolving  around  our  sun,  every  one  of  which 
may  be  inhabited  by  beings,  not  necessarily  with 
bodies  like  ours,  because  they  would  have  to  be 
adapted  to  their  own  physical  environment,  but 
perhaps  with  souls  like  ours.  Our  sun,  however, 
is  only  one  of  the  smaller  of  the  suns,  fixed  stars, 
apparently  without  number,  which  dot  our  heav- 
ens. In  some  parts  of  the  sky  there  appear  clus- 
ters of  them,  as  in  the  Milky  Way,  so  numerous 
that  they  appear  only  as  patches  of  light.  Some 
of  these  suns  are  so  far  away  that  it  takes  cen- 
turies for  the  light  from  them  to  reach  us.  A  few 
years  ago  a  star  of  very  large  magnitude  suddenly 
appeared  in  our  sky.  It  then  grew  gradually 
smaller  until  now  it  is  not  at  all  conspicuous.  As- 
tronomers told  us  that  this  new  star  really  rep- 
resented a  conflagration  in  the  heavens,  two  large 
suns  had  collided,  but  the  collision  had  taken 
place  over  one  hundred  years  before,  at  the  very 


112   The  Fourth  Dimension  and  the  Bible 

time  when  Napoleon  was  walking  the  world  like 
a  Colossus,  and  the  flash  of  light  caused  by  the 
impact  never  reached  earth  until  in  our  day.  In 
fact,  there  may  be  suns  so  far  from  us  and  cre- 
ated so  long  ago  that  light  from  them  has  not 
yet  reached  the  earth.  Each  of  these  countless 
suns  may  have  many  more  and  larger  planets 
revolving  about  them  than  is  the  case  with  our 
sun,  and  each  planet  may  be  inhabited  by  beings 
with  souls.  All  of  this  vast  infinitude  that  we 
have  been  contemplating  exists,  however,  simply 
in  our  space  of  three  dimensions.  There  may 
exist  an  infinite  number  of  other  three-dimen- 
sional spaces,  each  one  comprising  a  universe  as 
great  as  or  greater  than  our  own.  But  enfolding 
these  three-dimensional  spaces  as  in  a  garment 
may  be  a  four-dimensional  space,  and  that  may 
be  only  one  of  an  infinite  number  of  other  pos- 
sible four-dimensional  spaces.  Again,  these  four- 
dimensional  spaces  may  be  enfolded  by  five-di- 
mensional spaces;  and  so  on  thru  spaces  of  six, 
seven,  etc.,  dimensions  up  to  space  of  infinite  di- 
mensions, where  God  himself  dwells  subject  to  no 
space  limitations. 

But  now  man,  at  the  thought  of  these  possi- 
bilities, stands  aghast,  stupified;  he  exclaims,  "it 
can't  be  true,  it  is  all  too  vast,  too  stupendous  to 
lie   within   the   realms   of   the   possible."      Even 


Conclusion  113 

some  who  believe  in  God  are  inclined  to  think  that 
this  concept  exceeds  the  bounds  of  the  attainable. 
But  by  thus  placing  a  limit  to  what  God  can  do 
we  are  virtually  denying  that  he  is  God.  Instead 
of  being  restricted  to  those  possibilities  which 
may  be  humanly  conceived,  God  is  able  to  put  our 
unbelief  to  shame  by  exhibitions  of  power  of 
which  we  have  now  not  the  slightest  conception. 


INDEX 


Abbott,  41 

Abraham,   73 

Adam's  fall,  89 

Alcohols,  67 

Analogy,  17,  23,  76,  85 

An  Episode  in  Flatland,  41 

Angels,  45,  77 

Animal  kingdom,  64 

Annihilation   of  the   universe, 

93 
Anstadt,  107 
Apostles,   106 
Apparition,  43,  58 
Archangels,  45 
Astronomers,  80,  iii 
Atoms,  66 

Atonement,  Christ's,  89 
Automobile,   plane   section   of. 

Axis  of  rotation,  57,  63 

Beethoven,  33 

Bible,  17,  44,  45,  52,  73,  77,  81, 
89,    90,    91,   95>    100,    102, 
107,  108 
Biologist,  77 
Birth,  70,  82 
Blind,  33 

Blood  of  Jesus,  103 
Bodies,  celestial,  80 
Body  of  Jesus,  103 
Boundaries,  Flatland,  38 

Lineland,  42 

of  space  units,  22 
Bragdon,  72 


Carbon,  67 

Celestial  bodies,  80 

Chained   shadow   watchers   of 

Plato,  82 
Chemistry,  66 
Christ,  45,  52,  88,  106,  109 

atonement,  89 
Circle,  circumference  of,  12 
Cone,  Flatland  sections  of,  69, 

73,  96 
Concept  of  space,  iiflF. 
Configurations,    higher-dimen- 
sional, 79 

rotation  of  symmetrical,  54 
Conservation  of  energy,  81 

of  matter,  81 
Creation,  Ex-Nihilo,  31 

instantaneous,  97 

of  the  universe,  91 
Crucifixion  of  Christ,  45,  109 
Cruse,  widow's,  81 
Crystals,  68 
Cube,  19 

Flatland  sections  of  a,  75,  79 

motion   perpendicular  to    a, 
20,  48 
Curved  spaces,  12,  14,  15 
Cylinder,  Flatland  sections  of 
a,  91 

Daughter  of  Jairus,  100 
Dead,  raising  of  the,  99 
Deaf,  33 
Death,  70,  82 
Definition  of  space,  22 
Dextrose,  64 


"5 


ii6 


Index 


Dimensions  of  space,  iiff 

geometry  of  four,  29 

rotation,    reversal,   twist  in, 
61 

theorems  in  four,  58 

time  as  fourth,  71 
Distorted,  21,  63 
Dreams  and  dreamland,  72 

Earth,  first,  88 

Elements  of  the  tesseract,  21 
Elijah,  44,  45,  81,  loo 
Elisha,   100 

Embryos,  plane  sections  of,  78 
Empirical  Science,  29 
Energy,  conservation  of,  81 
Enoch,  44,  45,  81. 
Eternal,  73 
Eutychus,  100 
Evanescent,  76 
Evolution,  96 

Existence  of  higher  spaces,  23, 
25 

First  earth,  heaven,  88 
Fishes,  the  loaves  and,  81 
Five-dimensional  space,  16 
Flatland,   37,  48,   55,   69!?,   84, 

87 
boundaries,  38 
houses,  39 
life  in,  37ff 

sections  of  a  helix,  73 
sections  of  a  cone,  69,  73,  96 
sections  of  a  cube,  75,  79 
sections  of  a  cylinder,  91 

Flatlanders,  37ff,  SS^)  92 

Foot,  84 

Four-dimensional  spaces,  16 
space  unit,  20 

Four  dimensions,  geometry  of, 

life  in  space  of,  48 
rotation,    reversal,   twist   in, 

61 
theorems  in,  58 


Fourth  dimension,  time  as  the, 

7^ 
Freedom  of  motion  of  points, 

iiflF 

Generation  of  space  units,  i8ff 
Geographical      latitude      and 

longitude,   14 
Geometric  units  in  each  space, 

i8ff 
Geometry    not    the    same    as 

space,  28 
Geometry  of  four  dimensions, 

29,  58 
theorems  in  plane  and  solid, 

58 
Glass  globe,  49 
Grammatical  tenses,  73 
Grave,  loi 
Gravitation,  53 
Gruber,  31 
God,    31,    35,    44,    45,    73,    77, 

85,  88,  93,  95,  98,  102,  106, 

109 
Good  Friday,  100 
Gunnell,  26 

Heaven,  44,  86flF,  89 

description  of,  106 

first,  second,  third,  88 

of  heavens,  89 

recognition  in,  107 
Heels,  84 

Helix,  Flatland  sections  of,  73 
Hells,  86ff 
Helmholtz,  109 

Higher-dimensional  configura- 
tions 79 
Higher  spaces,  16 

existence  of,  23,  25 

divided     by     lower     spaces, 
23 
Hinton,  41 
Holy  City,  100 
Holy  Ghost,  109 
Hyper-spaces,  16 


Index 


117 


Illusory  and  real,  82 
Image  in  a  mirror,  61 

mental,  71 
Imagination,   28,   52 
Incomprehensible,  25,  26 
Infinite  number  of  points,  ii 

number   of   one-,   two-,    and 
three-dimensional     spaces, 

I2ff. 

space,  25 
Infinity,  25 

Inhabitants  of  other  spaces,  36 
Ink  stand,  14 
Instantaneous  creation,  97 
Isomeric  substances,  65 

Jairus,  daughter  of,  icxj 

Jericho,  81 

Jesus,  73,  8r,  100,  i02ff 

body  and  blood  of,  103 
Jews,  52 

Kant,  31 
Kelvin,  109 

Kingdoms,    animal    and   vege- 
table, 64 

Ladder,  16 

Latitude,  geographic,  14 

parallels  of,  14 
Laws,  natural,  108 
Lazarus,   looff 
Left-handed,  57 
Levulose,  64 
Life,    transitory   character   of, 

in  Flatland,  37 

in    four-dimensional     space, 

in  Lineland,  41 
Limits  of  space,  25 
Line,   motion   perpendicular  to 

a,  19,  47 
Lineland,  37!? 
life  in,  41 
boundaries,  42 


Linelanders,  375,  48,  54,  55 
Lines,  rotation  of  symmetrical, 

of  longitude,  14 
Loaves  and  fishes,  81 
Longitude,  geographical,  14 
Looking  glass  world,  62 
Lord's  Supper,  102 

Manna  in  the  wilderness,  81 
Manning,   31 
Mars,  message  from,  loi 
Mass  of  the  universe,  79 
Materialist,  no 
Mathematics,  pure,  29 
Matter,  conservation  of,  79 
Memory,  72 
Mental  image,  71 

vision,  71,  73 
Message  from  Mars,  loi 
Microscope,  78 
Milky  Way,  in 
Miracles,  40,  45,  108 
Mirror  image,  57,  61 
Molecules,  66 
Moses,  45 
Motion  of  points,  11  if 

perpendicular    to    a    line   or 
square,   19,  47 

perpendicular  to  a  cube,  20, 


Nain,  son  of  the  widow  of,  100 

N-dimensional  space,  16 

Napoleon,   112 

Natural  laws,  108 

Nebulae,  94 

Newcomb,  41,  48 

Number  of  points,  11 

Oil  in  the  widow's  cruse,  81 
One-dimensional  spaces,  iiff 
space  units,   18 

Parallels  of  latitude,  14 
Paul,  46,  100 


ii8 


Index 


Personality,  76 

Peter,  100 

Physics,  30 

Plane   geometry,   theorems   in, 

Plane  sections  of  an  automo- 
bile, 80 
of  an  embryo,  78 
Planets,  79,  94 
Plato,  82 

Points,  freedom  of  motion  of, 
I  iff 
moving      perpendicular      to 
spaces,  47ff 
Polarization  of  light,  64 
Prophets,  45,  108 
Pure  mathematics,  29 

Raising  of  the  dead,  99 
Real,  the  illusory  and  the,  82 
Recognition  in  heaven,  107 
Red  Sea,  81 

Rending  of  the  veil,  109 
Revelations  of  St.  John,  46 
Reversal  in  space  of  four  di- 
mensions, 61 
Right  handed,  57 
Robber,  49 

Rotation  of  symmetrical  lines, 
triangles,        tetrahedrons, 

.    54ff 

in  space  of  four  dimensions, 
61 
Rubber  band  or  ball,  63 

Sacrament  of  the  Lord's  Sup- 
per, 102 
Satan,  90 
Saul,  99 

Schubert,  59,  67 
Scientific  romances,  41 
Second  heaven,  88 
Sections  of  embryos,  78 
Sense  perceptions,  27 
Shadow  watchers  of  Plato,  82 
Solar  system,  25 


Soles,  84 

Solid    geometry,    theorems    in, 

58 
Son   of  the   Widow   of   Nain, 

100 
Soul,  99 
Space  defined,  22 

boundaries  of,  22 
Space,  zero-,  one-,  two-,  three-, 
four-,    five-,    higher-,    or 
hyper-,  iiff 
curved,  i2ff 

life  in  four-dimensional,  48 
limitations,  30 
limits,  25 
units,  i8ff 
Spaces   as   heavens   and   hells, 
86ff 
existence  of  higher,  25 
higher     spaces     divided    by 

lower,   23 
inhabitants  of  other,  36 
Spectrum,  34 
Sphere,  surface  of,  14,  48 
Spiritual,  35,^  53,  76,  103  ^ 
Square,    motion    perpendicular 

to  a,  19,  47 
Stars,  94,  III 
Suns,  79,  III 
Supernatural,  42,  53 
Swedenborg,  109 
Symbolic,  17,  20 
Symmetrical  configurations,  79 

rotation  of,  54,  56,  60 
Symmetry  in  animal  and  vege- 
table kingdoms,  64 


Tabitha,  100 
Table  top,  14 
Tartaric  acid,  67 
Taylor,  71 
Telegraph,  26 
Telephone,  26 

Temple,  rending  of  the  veil  of, 
109 


Index 


119 


Tenses,  grammatical,  73 
Tesseract,  20,  21 
Tetrahedrons,  59,  60,  65 
The    Fairyland   of    Geometry, 

41 

Theorems  in  plane,  solid  and 
four-dimensional  geome- 
try, 58 

Third  heaven,  88 

Thomas,  52 

Three-dimensional  spaces,  14 
space  units,  19 

Time  as  the  fourth  dimension, 

71 
Tire  tube,  64 
Transitory    character    of    life, 

76       ,       .     . 
Transubstantiation,       doctrine 

of,  103 

Triangles,  56,  57 

Twist  in  space  of  four  dimen- 
sions, 6 1 


Two-dimensional  spaces,  13 
unit,  19 
vision,  39 

Units  of  space,  i8ff 

boundaries  of  space,  22 

Universe,  annihilation  of,  93 
creation  of,  91 
mass  of,  79 

Veil   of  the  Temple,    rending 

of,  109 
Vision,  mental,  71,  73 
Visualizing,  28 

Wilderness,  manna  in  the,  81 
Widow's  cruse,  81 
Widow  of  Nain,  son  of,  100 
Wireless  telegraph,  26,  108 
telephone,   26 

Zero-dimensional  space,  11 
space  unit,  18 


DATE  DUE 

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